# On bundles of matrix pencils under strict equivalence111 This work has been
supported by the Agencia Estatal de Investigación of Spain through grants
PID2019-106362GB-I00 MCIN/ AEI/10.13039/501100011033/ and MTM2017-90682-REDT,
and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual
Agreement with UC3M in the line of Excellence of University Professors
(EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research
and Technological Innovation).

Fernando De Terán Departamento de Matemáticas, Universidad Carlos III de
Madrid, Avda. Universidad 30, 28911 Leganés, Spain<EMAIL_ADDRESS>Froilán M. Dopico Departamento de Matemáticas, Universidad Carlos III de
Madrid, Avda. Universidad 30, 28911 Leganés, Spain<EMAIL_ADDRESS>
###### Abstract

Bundles of matrix pencils (under strict equivalence) are sets of pencils
having the same Kronecker canonical form, up to the eigenvalues (namely, they
are an infinite union of orbits under strict equivalence). The notion of
bundle for matrix pencils was introduced in the 1990’s, following the same
notion for matrices under similarity, introduced by Arnold in 1971, and it has
been extensively used since then. Despite the amount of literature devoted to
describing the topology of bundles of matrix pencils, some relevant questions
remain still open in this context. For example, the following two: (a) provide
a characterization for the inclusion relation between the closures (in the
standard topology) of bundles; and (b) are the bundles open in their closure?
The main goal of this paper is providing an explicit answer to these two
questions. In order to get this answer, we also review and/or formalize some
notions and results already existing in the literature. We also prove that
bundles of matrices under similarity, as well as bundles of matrix polynomials
(defined as the set of $m\times n$ matrix polynomials of the same grade having
the same spectral information, up to the eigenvalues) are open in their
closure.

Keywords. Matrix, matrix pencil, matrix polynomial, spectral information,
strict equivalence, Kronecker canonical form, Jordan canonical form, orbit,
bundle, open set, closure, majorization.

AMS Subject Classification. 15A22, 15A18, 15A21, 15A54, 65F15.

##  1 Introduction

Orbits of matrices and matrix pencils arise as a natural object when dealing
with equivalence relations and their canonical forms. Orbits of matrices under
similarity were introduced by Arnold in [1], and in this case the canonical
form is the well-known Jordan Canonical Form (JCF). More precisely, the orbit
(under similarity) of a given matrix consists of all matrices with the same
JCF. In other words, it is the orbit of the given matrix (with size, say,
$n\times n$) under the action of similarity of the general linear group ${\rm
GL}_{n}({\mathbb{C}})$, consisting of all $n\times n$ invertible matrices with
complex entries, on the set of all $n\times n$ matrices with complex entries,
${\mathbb{C}}^{n\times n}$, namely

$\begin{array}[]{ccc}{\rm GL}_{n}({\mathbb{C}})\times{\mathbb{C}}^{n\times
n}&\rightarrow&{\mathbb{C}}^{n\times n}\\\ (P,A)&\mapsto&PAP^{-1}.\end{array}$

In the case of general (unstructured) matrix pencils (namely, pairs of
matrices of size $m\times n$), similarity is replaced by the so-called strict
equivalence, that is

$\begin{array}[]{ccc}{\rm GL}_{m}({\mathbb{C}})\times{\rm
GL}_{n}({\mathbb{C}})\times{\mathbb{C}}^{m\times n}\times{\mathbb{C}}^{m\times
n}&\rightarrow&{\mathbb{C}}^{m\times n}\times{\mathbb{C}}^{m\times n}\\\
(P,Q,A,B)&\mapsto&(PAQ,PBQ),\end{array}$

and the canonical form under this relation is the Kronecker Canonical Form
(KCF) (see [21, Ch. XII, §5]). Thus the orbit of a given pencil consists of
all pencils with the same KCF.

The notion of orbit has some limitations when studying the change of the
spectral information, because the eigenvalues of the matrix or the matrix
pencil must be fixed. But if, for instance, we are interested in analyzing the
change of the spectral information under small changes in the entries of the
matrices $A$ and $B$ above, we should allow the eigenvalues to change, since
this is what happens under generic perturbations. In order to overcome this
limitation, Arnold introduced the notion of bundle in his 1971 paper [1]. A
bundle is the union (usually infinite) of all orbits that have the same JCF
(or the same KCF for pencils) up to the specific values of the eigenvalues.
More precisely, and taking the words by Arnold for the case of matrices, “a
bundle is the set of all matrices whose Jordan normal forms differ only by
their eigenvalues, but for which the sets of distinct eigenvalues and the
orders of the Jordan blocks are the same”. For instance, for the matrix

$A=\left[\begin{array}[]{cc|c}\lambda_{0}&1&0\\\ 0&\lambda_{0}&0\\\ \hline\cr
0&0&\mu_{0}\end{array}\right],$

consisting of one Jordan block with size $2$ associated with the eigenvalue
$\lambda_{0}$ and one Jordan block with size $1$ associated with the
eigenvalue $\mu_{0}$, with $\lambda_{0}\neq\mu_{0}$, the corresponding bundle
is the set

${\cal B}(A):=\left\\{P\left[\begin{array}[]{cc|c}\lambda&1&0\\\
0&\lambda&0\\\ \hline\cr 0&0&\mu\end{array}\right]P^{-1}:\
\lambda,\mu\in{\mathbb{C}},\ \lambda\neq\mu,\
P\in\mbox{GL}_{3}({\mathbb{C}})\right\\}.$

The notion of bundle can be easily extended to matrix pencils, namely a bundle
is the set of matrix pencils with the same KCF, up to the eigenvalues. Bundles
of matrix pencils have been considered in many papers, mostly in the last $30$
years, like [7, 15, 17, 18, 19, 20, 29, 31]. Some of these references deal
with bundles of structured pencils, namely those enjoying some particular
symmetry in the coefficient matrices, $A$ and $B$, of the pencil $(A,B)$ (like
alternating, (skew-) Hermitian, (anti-)palindromic, or (skew-)symmetric, see,
for instance, [26] for the definition of all these structures), and in these
cases the strict equivalence relation is replaced by the congruence or
$*$-congruence relation. Even bundles of unstructured matrices under the
relation of congruence (or $*$-congruence) have been considered in the
literature, mostly arising from structured matrix pencils like, for instance,
in [9, 13, 14]. However, structured pencils are out of the analysis carried
out in the present work, and we consider only general (unstructured) matrix
pencils.

The canonical forms mentioned above (namely, the JCF for matrices or the KCF
for matrix pencils) are the representatives of the equivalence classes of
matrices or matrix pencils under the action of similarity or strict
equivalence (or congruence/$*$-congruence in the structured case),
respectively, (see, for instance, [18] for more information on these canonical
forms in the unstructured case, and [32] for structured matrix pencils). A
relevant question, not only theoretically, but also for applied and numerical
purposes (for instance, in the computation of the canonical forms) is to
determine which are the most “likely” representatives of these canonical
forms. Or, more precisely, given two different representatives of the
canonical form, to determine whether one of them is more “likely” than the
other or not. In order to answer this question, the standard approach, that
has been followed, for instance, in [7, 8, 10, 11, 12, 14, 15, 17, 18, 19, 29]
is to translate it to the context of orbits or, more in general, of bundles.
To be more precise, and focusing on strict equivalence of matrix pencils, each
bundle is associated with a particular pencil $L(\lambda)$ or, more in
general, with the KCF of $L(\lambda)$, which is determined up to the values of
the eigenvalues. Then, given two bundles ${\cal B}(L_{1})$ and ${\cal
B}(L_{2})$, we say that the KCF of $L_{1}(\lambda)$ is “more likely” (or “more
generic”) than the one of $L_{2}(\lambda)$ if the closure of the bundle of
$L_{2}(\lambda)$ is included in the closure of the bundle of $L_{1}(\lambda)$
(namely $\overline{\cal B}(L_{2})\subseteq\overline{\cal B}(L_{1})$, in the
notation that is used throughout the manuscript), where the closures are
considered in the standard topology of ${\mathbb{C}}^{m\times
n}\times{\mathbb{C}}^{m\times n}$, which is identified with
${\mathbb{C}}^{2mn}$. Therefore, the inclusion relations of bundle closures
determine the “likelihood” of the corresponding canonical forms.

For the relation of similarity of matrices (namely, the likelihood of the
JCF), a characterization of the inclusion relation between orbit closures is
known since the 1980’s [6, 28]. A characterization for the inclusion relation
between orbit closures of matrix pencils under strict equivalence was obtained
in [2] and [30], and for bundle closures of matrices under similarity, a
characterization for the inclusion relation is presented in [18, Th. 2.6].
However, we have been unable to find in the literature any explicit
characterization of the inclusion relation between bundle closures of matrix
pencils under strict equivalence, even though Theorem 3.3 in [18] provides an
explicit characterization of the so-called “covering” relation between bundle
closures of matrix pencils (where covering means that $\overline{\cal
B}(L_{2})\subseteq\overline{\cal B}(L_{1})$ and there is no any $L_{3}$ such
that $\overline{\cal B}(L_{2})\subset\overline{\cal
B}(L_{3})\subset\overline{\cal B}(L_{1})$, and $\subset$ means strict
inclusion).

In the arguments described at the end of the last-but-one paragraph, it is
implicitly assumed that the KCF of $L(\lambda)$ is the “generic” one in
$\overline{\cal B}(L)$. Let us recall that a subset, ${\cal S}_{0}$, of a
given set $\cal S$ (in a topological space) is called generic in $\cal S$ if
${\cal S}_{0}$ is open and dense in $\cal S$. Therefore, the assumption on the
genericity of the KCF just mentioned is equivalent to say that ${\cal B}(L)$
is open and dense in its closure. Clearly, ${\cal B}(L)$ is dense in
$\overline{\cal B}(L)$. The property of being open in its closure is well-
known for orbits of varieties under the action of a group (see, for instance,
[23, p. 60]). This is the case, for instance, of matrices under similarity and
matrix pencils under strict equivalence and congruence. However, we have not
found in the literature a corresponding result for bundles so, up to our
knowledge, the question on whether the bundles are open in their closure, is
still open.

Summarizing, the following two relevant questions arise:

  1. Q1.

To provide a characterization for the relation $\overline{\cal
B}(L_{2})\subseteq\overline{\cal B}(L_{1})$ to hold, for two given matrix
pencils $L_{1}$ and $L_{2}$.

  2. Q2.

To prove that bundles are open in their closure.

The main goal of the present work is to provide an answer to the previous two
questions. To get this answer, we revisit some notions that are already in the
literature, like the notion of “coalescence” of eigenvalues, as well as some
results, like Theorem 7.5 in [29], which is proven to be false.

As an aside result, we are also able to prove that bundles of matrices under
similarity, as well as bundles of matrix polynomials, defined as sets of
matrix polynomials with the same size and grade (see Section 4.1 for this
notion) having the same spectral information, up to the specific values of
their eigenvalue (see Section 4.2), are also open in their closure.

The rest of the paper is organized as follows. In Section 2 we introduce the
notation and basic notions used throughout the paper, together with some
already known results that are used later. Section 3 is devoted to the
solution of question Q1 above, whereas in Section 4 we provide an affirmative
answer to question Q2 for bundles of matrix pencils under strict equivalence,
as well as for matrices under similarity (in Section 4.1), and matrix
polynomials of higher degree (in Section 4.2). Finally, Section 5 presents a
summary of the main contributions of the paper, together with some lines of
further related research.

##  2 Basic definitions and notation

We use the following notation throughout the paper. By ${\mathbb{C}}^{m\times
n}$ we denote the set of $m\times n$ matrices with complex entries, whereas
${\rm GL}_{n}({\mathbb{C}})$ denotes the set of $n\times n$ invertible
matrices with complex entries. Also,
$\overline{\mathbb{C}}={\mathbb{C}}\cup\\{\infty\\}$. By $I_{k}$ we denote the
$k\times k$ identity matrix, and $\nu(A)$ denotes the dimension of the (right)
null space of the matrix $A$. Instead of using the representation of matrix
pencils as pairs of matrices of the same size, we will represent a matrix
pencil as $L(\lambda)=\lambda B+A$, with $A,B\in{\mathbb{C}}^{m\times n}$,
namely as a matrix polynomial of degree $1$ in the variable $\lambda$. We
often denote matrix pencils with a single capital letter (usually $L$ and $M$)
and, for the sake of simplicity, in the notation for notions associated with a
matrix pencil we omit the variable $\lambda$, and write just $L$ instead of
$L(\lambda)$.

The rank of the pencil $L(\lambda)$ (that is sometimes found in the literature
under the name “normal rank”), denoted by ${\mbox{\rm rank\,}}L$, is the rank
of $L(\lambda)$ considered as a matrix over the field of rational functions in
the variable $\lambda$. In other words, it is the size of the largest non-
identically zero minor of $L(\lambda)$.

Block-partitioned matrices (or matrix pencils) will appear frequently
throughout the manuscript, and the blocks indicated with $*$ are not relevant
in the arguments, developments, or results. For a block diagonal pencil (or
matrix) with diagonal blocks $A_{1},\ldots,A_{k}$ we use either the notation
${\mbox{\rm diag}}(A_{1},\ldots,A_{k})$ or $\bigoplus_{i=1}^{k}A_{i}$.

###  2.1 The KFC, orbits and bundles

Let us recall that the KCF of a matrix pencil $L(\lambda)$ (that we denote by
KCF($L$)) is a block diagonal pencil, whose diagonal (“canonical”) blocks can
be of the following four forms (see, for instance, [21, Ch. XII, §5]):

  * •

Jordan blocks associated with a finite eigenvalue $\mu$, namely $\lambda
I_{k}+J_{k}(\mu)$, for $k\geq 1$, where

$J_{k}(\mu):=\begin{bmatrix}-\mu&1\\\ &\ddots&\ddots\\\ &&-\mu&1\\\
&&&-\mu\end{bmatrix}_{k\times k}.$

  * •

Jordan blocks associated with the infinite eigenvalue, namely $\lambda
N_{k}+I_{k}$, for $k\geq 1$, where:

$N_{k}:=\begin{bmatrix}0&1\\\ &\ddots&\ddots&\\\ &&0&1\\\
&&&0\end{bmatrix}_{k\times k}.$

  * •

Right singular blocks, for $k\geq 0$:

$R_{k}(\lambda)=:\begin{bmatrix}\lambda&1\\\ &\lambda&1\\\ &&\ddots&\ddots\\\
&&&\lambda&1\end{bmatrix}_{k\times(k+1)}.$

  * •

Left singular blocks, $R_{k}(\lambda)^{\top}$, for $k\geq 0$.

The KCF of $L(\lambda)$ is determined up to permutation of the diagonal
blocks.

Let us note that $R_{0}(\lambda)$ is a null column, whereas
$R_{0}(\lambda)^{\top}$ is a null row. The KCF reveals all the invariants of a
matrix pencil under strict equivalence, namely the set of distinct finite and
infinite eigenvalues together with the number and the sizes of their
associated Jordan blocks, and the number and the sizes of the right and left
singular blocks.

Then, $\mu\in{\mathbb{C}}$ is a finite eigenvalue of $L(\lambda)$ if KCF($L$)
contains, at least, one Jordan block associated with $\mu$, and $L(\lambda)$
has the infinite eigenvalue if KCF($L$) contains, at least, one Jordan block
associated with the infinite eigenvalue. The pencil $L(\lambda)$ is said to be
regular if there are neither right nor left singular blocks in KCF($L$) (this
is equivalent to say that $L(\lambda)$ is square and $\det L(\lambda)$ is a
non-identically zero polynomial). By $\Lambda(L)$ we denote the spectrum of
the pencil $L(\lambda)$ (namely, the set of distinct eigenvalues of
$L(\lambda)$, both finite and infinite).

We denote by $W(\mu,L)=(W_{1}(\mu,L),W_{2}(\mu,L),\ldots)$ the Weyr
characteristic of the eigenvalue $\mu$ in the $m\times n$ matrix pencil
$L(\lambda)$. In other words, when $\mu\in{\mathbb{C}}$ (respectively,
$\mu=\infty$), $W_{i}(\mu,L)$, for $i\geq 1$, is the number of Jordan blocks
$\lambda I_{k}+J_{k}(\mu)$ (resp., $\lambda N_{k}+I_{k}$), with $k\geq i$, in
KCF($L$), (see, for instance, [4]). If $\mu$ is not an eigenvalue of
$L(\lambda)$, then $W(\mu,L)=(0,0,\ldots)$. Also,
$r(L)=(r_{0}(L),r_{1}(L),\ldots)$ and
$\ell(L)=(\ell_{0}(L),\ell_{1}(L),\ldots)$ denote, respectively, the Weyr
characteristic of the right and left singular structure. In other words,
$r_{i}(L)$ (respectively, $\ell_{i}(L)$) is the number of right (resp., left)
singular blocks $R_{k}(\lambda)$ of size $k\times(k+1)$ (resp.,
$R_{k}(\lambda)^{\top}$ of size $(k+1)\times k$), with $k\geq i$, in KCF($L$).
In particular, $r_{0}(L)=n-{\mbox{\rm rank\,}}L$ and $\ell_{0}(L)=m-{\mbox{\rm
rank\,}}L$. Note that $W(\mu,L),r(L),$ and $\ell(L)$ are lists of non-
increasing integers. We use the notation ${\cal L}_{1}\prec{\cal L}_{2}$ to
denote the majorization of two lists of non-increasing integers, namely:
$\sum_{i=1}^{j}{\cal L}_{1}(i)\leq\sum_{i=1}^{j}{\cal L}_{2}(i)$, for all
$j\geq 1$, assuming that the lists start with $i=1$.

For $\mu\in{\mathbb{C}}$ and $\varepsilon>0$, the $\varepsilon$-neighborhood
of $\mu$ is defined as $B(\mu,\varepsilon):=\\{z\in\overline{\mathbb{C}}:\
|z-\mu|<\varepsilon\\}$, whereas if $\mu=\infty$, we set
$B(\infty,\varepsilon):=\\{z\in{\mathbb{C}}:\ |z|>\varepsilon^{-1}\\}$).

Following the notation in [29], $\Phi$ denotes the set of all one-to-one
mappings of $\overline{\mathbb{C}}$ to itself. We also use the notation $\Psi$
for the set of all mappings from $\overline{\mathbb{C}}$ to itself (not
necessarily one-to-one). Then, if $K_{L}(\lambda):={\rm KCF}(L)$ (where the
canonical blocks are given in any order), and $\psi\in\Psi$, we denote by
$\psi(L)$ any pencil which is strictly equivalent to the pencil obtained from
$K_{L}(\lambda)$ after replacing the Jordan blocks associated with the
eigenvalue $\mu\in\overline{\mathbb{C}}$ by Jordan blocks of the same size
associated with the eigenvalue $\psi(\mu)$, for any eigenvalue $\mu$ of
$K_{L}(\lambda)$. Note that this pencil $\psi(L)$ is not uniquely determined,
but all pencils $\psi(L)$ are strictly equivalent to each other.

For a given matrix pencil $L(\lambda)=\lambda B+A$, with
$A,B\in{\mathbb{C}}^{m\times n}$, we set:

$\begin{array}[]{ccll}{\cal O}(L)&:=&\\{\lambda PBQ+PAQ\,:\ P\in\mbox{\rm
GL}_{m}({\mathbb{C}}),Q\in\mbox{GL}_{n}({\mathbb{C}})\\}&\mbox{(orbit of
$L(\lambda)$),}\\\ {\cal B}(L)&:=&\displaystyle\bigcup_{\varphi\in\Phi}{\cal
O}(\varphi(L))&\mbox{(bundle of $L(\lambda)$).}\end{array}$

Note that ${\cal O}(\varphi(L))$ is well defined, regardless of the particular
pencil $\varphi(L)$, since, as mentioned above, all pencils $\varphi(L)$ are
strictly equivalent.

By definition, bundles are the union of orbits of all matrix pencils having
the same KCF up to the eigenvalues. Note that this union is infinite provided
that the pencils have, at least, one eigenvalue. However, if KCF($L$) contains
only blocks of the form $R_{k}(\lambda)$ and/or $R_{k}(\lambda)^{\top}$, then
${\cal B}(L)={\cal O}(L)$. Some special attention should be paid to the
infinite eigenvalue. More precisely, the reason for considering maps over
$\overline{\mathbb{C}}$ is, precisely, to include the infinite eigenvalue in
the bundles. For instance, if $L(\lambda)$ is of the form

$L(\lambda)=\lambda\left[\begin{array}[]{cc|c}1&0&0\\\ 0&1&0\\\ \hline\cr
0&0&1\end{array}\right]+\left[\begin{array}[]{cc|c}-\mu&1&0\\\ 0&-\mu&0\\\
\hline\cr 0&0&-\widetilde{\mu}\end{array}\right],\qquad\mu\neq\widetilde{\mu}$

(namely, the pencil has two different finite eigenvalues $\mu$ and
$\widetilde{\mu}$ with Jordan blocks of sizes $2$ and $1$, respectively) then

$\begin{array}[]{ccl}{\cal B}(L)&=&\footnotesize\left\\{\lambda
PQ+P\left[\begin{array}[]{cc|c}-a&1&\\\ 0&-a&\\\
\hline\cr&&-\widetilde{a}\end{array}\right]Q:a,\widetilde{a}\in{\mathbb{C}},a\neq\widetilde{a},\
P,Q\in\mbox{GL}_{3}({\mathbb{C}})\right\\}\\\
&&\footnotesize\bigcup\left\\{\lambda P\left[\begin{array}[]{cc|c}1&0&\\\
0&1&\\\ \hline\cr&&0\end{array}\right]Q+P\left[\begin{array}[]{cc|c}-a&1&\\\
0&-a&\\\ \hline\cr&&1\end{array}\right]Q:a\in{\mathbb{C}},\
P,Q\in\mbox{GL}_{3}({\mathbb{C}})\right\\}\\\
&&\footnotesize\bigcup\left\\{\lambda P\left[\begin{array}[]{cc|c}0&1&\\\
0&0&\\\ \hline\cr&&1\end{array}\right]Q+P\left[\begin{array}[]{cc|c}1&0&\\\
0&1&\\\ \hline\cr&&-a\end{array}\right]Q:a\in{\mathbb{C}},\
P,Q\in\mbox{GL}_{3}({\mathbb{C}})\right\\}.\end{array}$

The first set in the union above corresponds to a direct sum of two Jordan
blocks associated with a couple of finite (different) eigenvalues, the second
one corresponds to a Jordan block of size $2$ associated with a finite
eigenvalue, together with a Jordan block of size $1$ associated with the
infinite eigenvalue, and the third set corresponds to a Jordan block of size
$2$ associated with the infinite eigenvalue, together with a Jordan block of
size $1$ associated with a finite eigenvalue.

We will use the standard notation $\overline{S}$ for the closure, in the
standard topology, of the set $S$. In this context, the set of matrix pencils
$\lambda B+A$, with $A,B\in{\mathbb{C}}^{m\times n}$, is identified with
${\mathbb{C}}^{2mn}$, and we consider the standard topology in this set. For
the closure of the orbit and the bundle of a matrix pencil $L(\lambda)$ we use
the notation $\overline{{\cal O}}(L)$ and $\overline{{\cal B}}(L)$,
respectively.

###  2.2 Coalescence of eigenvalues

The notion of coalescence of eigenvalues, which is key to describe the
inclusion relationships for closures of bundles of matrices and matrix
pencils, has been used in previous references, including [17, 18]. We state
here a formal definition, which is equivalent to the one mentioned in [18, Th.
3.3-(5)]. We recall that the union of two lists of non-increasing integers
(like the Weyr characteristics), say ${\cal L}_{1}$ and ${\cal L}_{2}$, that
we denote by ${\cal L}_{1}\cup{\cal L}_{2}$, consists of a new list of non-
increasing integers which is obtained by arranging all numbers in ${\cal
L}_{1}$ and ${\cal L}_{2}$ in a non-increasing order. For instance, if ${\cal
L}_{1}=(5,2)$ and ${\cal L}_{2}=(6,3,3,2,1),$ then ${\cal L}_{1}\cup{\cal
L}_{2}=(6,5,3,3,2,2,1)$.

To understand the following definition, we recall that $\Psi$ denotes the set
of mappings from $\overline{\mathbb{C}}$ to itself.

######  Definition 1.

(Coalescence of eigenvalues). Let $L(\lambda)$ be a matrix pencil with
distinct eigenvalues $\mu_{1},\ldots,\mu_{s}$, and let $\psi\in\Psi$. Then,
$\psi_{c}(L)$ is any matrix pencil of the same size as $L(\lambda)$ satisfying
the following three properties:

  * •

$r(\psi_{c}(L))=r(L)$,

  * •

$\ell(\psi_{c}(L))=\ell(L)$, and

  * •

$W(\mu,\psi_{c}(L))=\bigcup_{\mu_{i}\in\psi^{-1}(\mu)}W(\mu_{i},L)$, for all
$\mu\in\overline{{\mathbb{C}}}$.

We say that the eigenvalues $\mu_{i_{1}},\ldots,\mu_{i_{d}}$ of $L(\lambda)$
have coalesced to the eigenvalue $\mu$ in $\psi_{c}(L)$ if
$\psi^{-1}(\mu)=\\{\mu_{i_{1}},\ldots,\mu_{i_{d}}\\}\cup S$, with
$S\cap\Lambda(L)=\emptyset$.

######  Remark 2.

The matrix pencil $\psi_{c}(L)$ in Definition 1 is not uniquely defined, but
all pencils $\psi_{c}(L)$, for some given $L(\lambda)$ and $\psi$, are
strictly equivalent to each other, since they all have the same KCF (as it
happens with $\psi(L)$). Moreover, note that ${\cal B}(\psi_{c}(L))={\cal
B}(\widetilde{\psi}_{c}(L))$, for any $\psi,\widetilde{\psi}\in\Psi$ such that
$\psi(\mu)=\psi(\widetilde{\mu})$ if and only if
$\widetilde{\psi}(\mu)=\widetilde{\psi}(\widetilde{\mu})$, for any
$\mu\neq\widetilde{\mu}$.

Coalescence of eigenvalues is a way of “gathering” eigenvalues by taking the
union of their Weyr characteristics. The following example aims to illustrate
this notion.

######  Example 3.

Let $L(\lambda)$ be the following pencil, already given in KCF:

$\begin{array}[]{rl}L(\lambda)={\mbox{\rm
diag}}&(R_{3}(\lambda),R_{1}(\lambda),\lambda I_{2}+J_{2}(0),\lambda
I_{2}+J_{2}(0),\lambda I_{1}+J_{1}(0),\\\ &\lambda I_{3}+J_{3}(1),\lambda
I_{2}+J_{2}(1),\lambda I_{4}+J_{4}(2),R_{2}(\lambda)^{\top}),\end{array}$

so that $r(L)=(2,2,1,1),$ $\ell(L)=(1,1,1),$ and $W(0,L)=(3,2),$
$W(1,L)=(2,2,1),$ $W(2,L)=(1,1,1,1).$

Let $\psi:\overline{{\mathbb{C}}}\rightarrow\overline{\mathbb{C}}$ be such
that $\psi(0)=\psi(1)=\psi(2)=1$. Then any pencil $\psi_{c}(L)$ is of the
form:

$\psi_{c}(L)=P\cdot{\mbox{\rm diag}}(R_{3}(\lambda),R_{1}(\lambda),\lambda
I_{9}+J_{9}(1),\lambda I_{4}+J_{4}(1),\lambda
I_{1}+J_{1}(1),R_{2}(\lambda)^{\top})\cdot Q,$

for some invertible matrices $P,Q$. Note, indeed, that, for any $\psi_{c}(L)$
as above, $r(\psi_{c}(L))=(2,2,1,1)=r(L)$,
$\ell(\psi_{c}(L))=(1,1,1)=\ell(L)$, and
$W(1,\psi_{c}(L))=(3,2,2,2,1,1,1,1,1)=W(0,L)\cup W(1,L)\cup W(2,L)$.

However, if $\psi$ is such that $\psi(0)=\psi(2)=1,\psi(1)=5$, then any pencil
$\psi_{c}(L)$ is now of the form

$\begin{array}[]{cl}\psi_{c}(L)=&P\cdot{\mbox{\rm
diag}}(R_{3}(\lambda),R_{1}(\lambda),\lambda I_{6}+J_{6}(1),\lambda
I_{2}+J_{2}(1),\lambda I_{1}+J_{1}(1),\\\ &\lambda I_{3}+J_{3}(5),\lambda
I_{2}+J_{2}(5),R_{2}(\lambda)^{\top})\cdot Q,\end{array}$

for some invertible matrices $P,Q$. Note, indeed, that, for any $\psi_{c}(L)$
as above, $r(\psi_{c}(L))=(2,2,1,1)=r(L)$,
$\ell(\psi_{c}(L))=(1,1,1)=\ell(L)$, and
$W(1,\psi_{c}(L))=(3,2,1,1,1,1)=W(0,L)\cup W(2,L)$,
$W(5,\psi_{c}(L))=(2,2,1)=W(1,L)$.

###  2.3 Some basic results

We are interested in the majorization of the Weyr characteristics of an
eigenvalue in two given matrix pencils. This majorization is defined by
inequalities on the sum of the first elements of the corresponding Weyr
characteristic, so a formula for this sum would be quite useful to this end.
Such a formula will come from the rank of certain big block-partitioned
matrices. For describing this, we follow some of the developments in the PhD.
thesis [3], that we include here for the sake of completeness.

Given the matrix pencil $L(\lambda)=\lambda B+A$, with
$A,B\in{\mathbb{C}}^{m\times n}$ and $\mu\in{\mathbb{C}}$, we define the
following block-partitioned matrices with $d$ block columns and $d$ block
rows:

$P_{\mu}^{d}(L):=\begin{bmatrix}L(\mu)&0&\ldots&\ldots&0\\\
B&L(\mu)&\ddots&&\vdots&\\\ 0&B&\ddots&\ddots&\vdots&\\\
\vdots&\ddots&\ddots&L(\mu)&0\\\ 0&\ldots&0&B&L(\mu)\end{bmatrix}_{dm\times
dn},\qquad\mbox{for $d\geq 1$}.$ (1)

Note that

$P_{\mu}^{d}(L)=\begin{bmatrix}0&0\\\ I_{d-1}&0\end{bmatrix}\otimes
B+I_{d}\otimes L(\mu),$

where $\otimes$ denotes the Kronecker product.

For $d_{1},\ldots,d_{s}\geq 1$ and
$\lambda_{1},\ldots,\lambda_{s}\in{\mathbb{C}}$ (different to each other), we
define the following block-partitioned matrices:

$P_{\lambda_{1},\ldots,\lambda_{s}}^{d_{1},\ldots,d_{s}}(L):=\begin{bmatrix}P_{\lambda_{1}}^{d_{1}}(L)&0&0&\ldots&0\\\
Q_{d_{2},d_{1}}(B)&P_{\lambda_{2}}^{d_{2}}(L)&0&\ldots&0\\\
\vdots&\ddots&\ddots&&\vdots\\\
0&0&\ldots&Q_{d_{s},d_{s-1}}(B)&P_{\lambda_{s}}^{d_{s}}(L)\end{bmatrix},$

where, for $2\leq i\leq s$,

$Q_{d_{i},d_{i-1}}(B):=\begin{bmatrix}0&\ldots&0&B\\\ 0&\ldots&0&0\\\
\vdots&\ddots&\vdots&\vdots&\\\
0&\ldots&0&0\end{bmatrix}\in{\mathbb{C}}^{d_{i}m\times d_{i-1}n}.$

######  Lemma 4.

Let $L(\lambda)=\lambda B+A$. Then, for any distinct
$\lambda_{1},\ldots,\lambda_{s}\in{\mathbb{C}}$ and $d_{1},\ldots,d_{s}\geq
1$,

$\nu({\mbox{\rm
diag}}(P_{\lambda_{1}}^{d_{1}}(L),\ldots,P_{\lambda_{s}}^{d_{s}}(L)))=\nu(P_{\lambda_{1},\ldots,\lambda_{s}}^{d_{1},\ldots,d_{s}}(L)),$

where $\nu(Z)$ denotes the dimension of the right nullspace of the matrix $Z$.

###### Proof.

The result is a consequence of the fact that
$P_{\lambda_{1},\ldots,\lambda_{s}}^{d_{1},\ldots,d_{s}}(L)$ is equivalent, by
elementary block-row and block-column operations, to the block diagonal matrix
${\mbox{\rm
diag}}(P_{\lambda_{1}}^{d_{1}}(L),\ldots,P_{\lambda_{s}}^{d_{s}}(L))$ (because
these operations preserve the rank and, as a consequence, the dimension of the
null space, of the matrix). To see this equivalence, let us first consider
just two big blocks, namely $s=2$, so the matrix reads
$P_{\alpha,\beta}^{d_{1},d_{2}}(L)$, with $\alpha,\beta\in{\mathbb{C}}$ and
$\alpha\neq\beta$. Note that, for any $s\in{\mathbb{C}}$,

$sB=s\left(\frac{1}{\alpha-\beta}L(\alpha)-\frac{1}{\alpha-\beta}L(\beta)\right).$
(2)

Therefore, adding to the $d_{1}$th block-column of
$P_{\alpha,\beta}^{d_{1},d_{2}}(L)$ the $(d_{1}+1)$st block column multiplied
by $1/(\alpha-\beta)$, and to the $(d_{1}+1)$st block-row the $d_{1}$th one
multiplied by $-1/(\alpha-\beta)$, the resulting matrix has a $0$ block in the
$(d_{1}+1,d_{1})$ block position, instead of the block $B$ that appears in
$P_{\alpha,\beta}^{d_{1},d_{2}}(L)$, but the blocks in the positions
$(d_{1}+1,d_{1}-1)$ and $(d_{1}+2,d_{1})$ are of the form $s_{1}B$ and
$s_{2}B$, for some $s_{1},s_{2}\in{\mathbb{C}}$. These blocks are in a block
diagonal that is below the one containing $B$ in the original matrix
$P_{\alpha,\beta}^{d_{1},d_{2}}(L)$, so the previous block row and block
column operations have taken the block $B$ to some multiples of $B$ in the
lower block diagonal. Because of (2), we can find appropriate elementary block
row and block column operations that turn these blocks into $0$, but creates
some nonzero blocks, which are, again, multiples of $B$, in the next lower
block diagonal. Proceeding recursively in this way, we end up with a matrix
which is equal to ${\mbox{\rm
diag}}(P_{\alpha}^{d_{1}}(L),P_{\beta}^{d_{2}}(L)$ except for a nonzero block
of the form $sB$, for some $s\in{\mathbb{C}}$, in the $(d_{1}+d_{2},1)$ block-
position. Using again (2), an appropriate linear combination of the first
block row and the last (i. e., the $(d_{1}+d_{2})$th one) block column will
shrink this block to $0$, so we end up with ${\mbox{\rm
diag}}(P_{\alpha}^{d_{1}}(L),P_{\beta}^{d_{2}}(L))$, as wanted.

If there are more than two blocks, we partition the matrix
$P_{\lambda_{1},\ldots,\lambda_{s}}^{d_{1},\ldots,d_{s}}(L)$ into a $2\times
2$ block matrix, as follows:

$P_{\lambda_{1},\ldots,\lambda_{s}}^{d_{1},\ldots,d_{s}}(L)=\begin{bmatrix}P_{\lambda_{1},\ldots,\lambda_{s-1}}^{d_{1},\ldots,d_{s-1}}(L)&0\\\
Q_{d_{s},\widetilde{d}_{s-1}}(B)&P_{\lambda_{s}}^{d_{s}}(L)\end{bmatrix},$ (3)

with $\widetilde{d}_{s-1}:=d_{1}+d_{2}+\cdots+d_{s-1}$. Using appropriate
elementary block-row and block-column operations as explained before for the
case $s=2$, this matrix is equivalent to

$\begin{bmatrix}P_{\lambda_{1},\ldots,\lambda_{s-1}}^{d_{1},\ldots,d_{s-1}}(L)&0\\\
0&P_{\lambda_{s}}^{d_{s}}(L)\end{bmatrix}.$ (4)

To be more precise, in this case, instead of (2), we need to use the identity

$sB=s\left(\frac{1}{\lambda_{i}-\lambda_{s}}L(\lambda_{i})-\frac{1}{\lambda_{i}-\lambda_{s}}L(\lambda_{s})\right),$

for $i=1,\ldots,s-1$, and the block-row and block-column operations produce
some multiples of $B$ in the $(2,1)$ block of (3), that, following the same
procedure as before, will move from one block diagonal to the “lower” one.
Then, after a finite number of elementary block-row and block-column
operations we arrive at (4). From (4) we can proceed in the same way with the
upper left block
$P_{\lambda_{1},\ldots,\lambda_{s-1}}^{d_{1},\ldots,d_{s-1}}(L)$, and so on,
until we get ${\mbox{\rm
diag}}(P_{\lambda_{1}}^{d_{1}}(L),\ldots,P_{\lambda_{s}}^{d_{s}}(L))$. ∎

The following result provides a formula for the sum of the first $d$ terms in
the Weyr characteristic of an eigenvalue of a matrix pencil, in terms of the
dimension of the null space of the matrix in (1).

######  Lemma 5.

Let $L(\lambda)$ be an $m\times n$ matrix pencil and $\mu\in{\mathbb{C}}$.
Then, for all $d\geq 1$:

$\nu(P_{\mu}^{d}(L))=\sum_{i=1}^{d}W_{i}(\mu,L)+d\,r_{0}(L),$

where $r_{0}(L)=n-{\mbox{\rm rank\,}}L$.

###### Proof.

It is straightforward to see that, if $\widetilde{L}(\lambda)$ is a strictly
equivalent pencil to $L(\lambda)$, then
$\nu(P_{\mu}^{d}(L))=\nu(P_{\mu}^{d}(\widetilde{L}))$. Therefore, we may
assume that $L(\lambda)$ is given in KCF. Let us decompose
$L(\lambda)={\mbox{\rm diag}}(L_{r}(\lambda),L_{\rm
reg}(\lambda),L_{\ell}(\lambda))$, where $L_{\rm reg}(\lambda)$ contains all
Jordan blocks of $L(\lambda)$, $L_{r}(\lambda)$ contains all blocks of the
form $R_{k}(\lambda)$, whereas $L_{\ell}(\lambda)$ contains the blocks of the
form $R_{k}(\lambda)^{\top}$. By means of column and row permutations,
$P_{\mu}^{d}({\mbox{\rm diag}}(L_{r},L_{\rm reg},L_{\ell}))$ is strictly
equivalent to ${\mbox{\rm diag}}(P_{\mu}^{d}(L_{r}),P_{\mu}^{d}(L_{\rm
reg})),P_{\mu}^{d}(L_{\ell}))$, so
$\nu(P_{\mu}^{d}(L))=\nu(P_{\mu}^{d}({\mbox{\rm diag}}(L_{r},L_{\rm
reg},L_{\ell})))=\nu({\mbox{\rm diag}}(P_{\mu}^{d}(L_{r}),P_{\mu}^{d}(L_{\rm
reg}),P_{\mu}^{d}(L_{\ell})))=\nu(P_{\mu}^{d}(L_{r}))+\nu(P_{\mu}^{d}(L_{\rm
reg}))+\nu(P_{\mu}^{d}(L_{\ell}))$.

Now, since $P_{\mu}^{d}(L_{\ell})$ has full column rank, for every
$\mu\in{\mathbb{C}}$, then $\nu(P_{\mu}^{d}(L_{\ell}))=0$. Also, since
$r_{0}(L)$ is the number of blocks $R_{k}(\lambda)$ in $L_{r}(\lambda)$, it is
also straightforward to see that $\nu(P_{\mu}^{d}(L_{r}))=d\,r_{0}(L)$.
Finally, $\nu(P_{\mu}^{d}(L_{\rm reg}))=\sum_{i=1}^{d}W_{i}(\mu,L_{\rm
reg})=\sum_{i=1}^{d}W_{i}(\mu,L)$, where the first identity is also
straightforward to get (see also [22, p. 36]). ∎

In the case where $r_{0}=0$, Lemma 5 is a consequence of the developments
carried out in [24, §5].

The Weyr characteristic of the eigenvalue $\mu$ in the pencil $L(\lambda)$ is
the conjugate partition of the Segre characteristic, denoted by
$S(\mu,L)=(S_{1}(\mu,L),S_{2}(\mu,L),\ldots)$, where $S_{i}(\mu,L)$ is the
size of the $i$th largest Jordan block associated with the eigenvalue $\mu$ in
KCF($L$) (see, for instance, [4]). In general, the conjugate of a partition,
${\bf a}$, is the partition, denoted by ${\bf a}^{\sharp}$, whose $i$th
element, for $i\geq 1$, is equal to the number of elements in ${\bf a}$ which
are greater than or equal to $i$. Also, the sum of a finite number of
partitions is the partition whose $i$th element is the sum of the $i$th
elements in all partitions (adding zeroes at the end of the partitions if
necessary). The following result can be found in [25, p. 6].

######  Lemma 6.

Let ${\bf a}_{1},\ldots,{\bf a}_{k}$ be partitions. Then

$\left(\bigcup_{i=1}^{k}{\bf a}_{i}\right)^{\sharp}=\sum_{i=1}^{k}{\bf
a}_{i}^{\sharp},\qquad\mbox{and}\qquad\left(\sum_{i=1}^{k}{\bf
a}_{i}\right)^{\sharp}=\bigcup_{i=1}^{k}{\bf a}_{i}^{\sharp}.$

The characterization for the inclusion between orbit closures of matrix
pencils, which was obtained independently in [2] and [30], and reformulated in
[4], will be used several times in the paper. We include it here for
completeness (the statement we present is in between the ones of [4, Lemma
1.3] and [18, Th. 3.1]).

######  Theorem 7.

(Characterization of the inclusion between orbit closures). The matrix pencil
$P_{2}(\lambda)$ belongs to the closure of the orbit of the pencil
$P_{1}(\lambda)$ (in other words, $\overline{\cal
O}(P_{2})\subseteq\overline{\cal O}(P_{1})$) if and only if the following
relations hold:

  * (i)

$r(P_{2})\prec r(P_{1})+(h,h,\ldots),$

  * (ii)

$\ell(P_{2})\prec\ell(P_{1})+(h,h,\ldots),$

  * (iii)

$W(\mu,P_{1})\prec W(\mu,P_{2})+(h,h,\ldots),$ for all
$\mu\in\overline{\mathbb{C}}$,

where $h:={\mbox{\rm rank\,}}P_{1}-{\mbox{\rm rank\,}}P_{2}$.

One of the main goals of this paper is to provide a characterization like the
one in Theorem 7 for closures of bundles instead of orbits. This will be
provided in Theorem 12.

##  3 A characterization for the inclusion of bundle closures of matrix
pencils

The main result of this section is Theorem 12, which provides a
characterization for the inclusion of bundle closures of matrix pencils. To
prove it we use the following technical result. It is a consequence of [3,
Teorema 2.3], adapted to the conditions in the statement. In the proof we use
the notion of reversal of the matrix pencil $L(\lambda)=\lambda B+A$, defined
as ${\rm rev}L:=\lambda A+B$.

######  Theorem 8.

Let $\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ be a sequence of $m\times n$
complex matrix pencils such that:

  1. (i)

$M_{k}(\lambda)\in{\cal B}(L)$, for some $m\times n$ complex matrix pencil
$L(\lambda)$ and for all $k\in\mathbb{N}$,

  2. (ii)

$\lambda_{1,k},\ldots,\lambda_{s,k}\in\overline{\mathbb{C}}$ are distinct
eigenvalues of $M_{k}(\lambda)$, for $k\in\mathbb{N}$, and
$W(\lambda_{i,k_{1}},M_{k_{1}})=W(\lambda_{i,k_{2}},M_{k_{2}})$, for all
$i=1,\ldots,s$, and all $k_{1},k_{2}\in\mathbb{N}$,

  3. (iii)

$\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ converges to $M(\lambda)$, and

  4. (iv)

the sequence $\\{\lambda_{i,k}\\}_{k\in\mathbb{N}}$ converges to $\mu$, for
all $i=1,\ldots,s$, where $\mu\in\overline{\mathbb{C}}$.

Then $\bigcup_{i=1}^{s}W(\lambda_{i,k},M_{k})\prec W(\mu,M)+(h,h,\ldots)$,
where $h:={\mbox{\rm rank\,}}L-{\mbox{\rm rank\,}}M$.

###### Proof.

Let us first assume that $\mu\in{\mathbb{C}}$ (namely, $\mu\neq\infty$). Then,
for $k$ large enough, $\lambda_{i,k}\in{\mathbb{C}}$, for all $1\leq i\leq s$.
Therefore, we may assume that $\lambda_{i,k}\in{\mathbb{C}}$, for all $1\leq
i\leq s$ and all $k\in\mathbb{N}$. In the rest of this case, we essentially
follow the proof of [3, Teorema 2.3].

Set $m=(m_{1},m_{2},\ldots):=\bigcup_{i=1}^{s}W(\lambda_{i,k},M_{k})$ (note
that this does not depend on $k$, by condition (ii) in the statement). By
definition of union of Weyr characteristics, for each $d\geq 1$, there are
$d_{1},\ldots,d_{s}\geq 0$ such that $d_{1}+\cdots+d_{s}=d$ and

$\sum_{i=1}^{d}m_{i}=\sum_{j=1}^{d_{1}}W_{j}(\lambda_{1,k},M_{k})+\cdots+\sum_{j=1}^{d_{s}}W_{j}(\lambda_{s,k},M_{k}),$
(5)

for all $k\in\mathbb{N}$, where $\sum_{j=1}^{0}W_{j}(\lambda_{i,k},M_{k}):=0$,
for $i=1,\ldots,s$.

Since the pencils $M_{k}(\lambda)$ converge to $M(\lambda)$ and the values
$\lambda_{i,k}$ converge to $\mu$, for all $1\leq i\leq s$, taking into
account that $d_{1}+\cdots+d_{s}=d$, we conclude that the matrices
$P_{\lambda_{1,k},\ldots,\lambda_{s,k}}^{d_{1},\ldots,d_{s}}(M_{k})$ (where,
if $d_{i}=0$ for some $1\leq i\leq s$, the block row and block column
corresponding to $P_{\lambda_{i,k}}^{d_{i}}(M_{k})$ is not present) converge
to $P_{\mu}^{d}(M)$. Then, by the lower semi-continuity of the rank, we get

${\mbox{\rm rank\,}}P_{\mu}^{d}(M)\leq{\mbox{\rm
rank\,}}P_{\lambda_{1,k},\ldots,\lambda_{s,k}}^{d_{1},\ldots,d_{s}}(M_{k}),$
(6)

for $k$ large enough.

From Lemma 5, and taking into account that ${\mbox{\rm
rank\,}}M_{k}={\mbox{\rm rank\,}}L$, for all $k\in\mathbb{N}$, we get, for
each $1\leq i\leq s$ and all $k\in\mathbb{N}$:

$\sum_{j=1}^{d_{i}}W_{j}(\lambda_{i,k},M_{k})+d_{i}(n-{\mbox{\rm
rank\,}}L)=\nu(P_{\lambda_{i,k}}^{d_{i}}(M_{k}))=nd_{i}-{\mbox{\rm
rank\,}}(P_{\lambda_{i,k}}^{d_{i}}(M_{k})).$

From this identity and (5) we obtain, for all $k\in\mathbb{N}$,

$\begin{array}[]{ccl}\displaystyle\sum_{i=1}^{d}m_{i}+d(n-{\mbox{\rm
rank\,}}L)&=&\displaystyle\sum_{i=1}^{s}\left(\sum_{j=1}^{d_{i}}W_{j}(\lambda_{i,k},M_{k})+d_{i}(n-{\mbox{\rm
rank\,}}L)\right)\\\ &=&\displaystyle nd-\sum_{i=1}^{s}{\mbox{\rm
rank\,}}(P_{\lambda_{i,k}}^{d_{i}}(M_{k}))\\\ &=&nd-{\mbox{\rm
rank\,}}{\mbox{\rm
diag}}(P_{\lambda_{1,k}}^{d_{1}}(M_{k}),\ldots,P_{\lambda_{s,k}}^{d_{s}}(M_{k}))\\\
&=&\nu({\mbox{\rm
diag}}(P_{\lambda_{1,k}}^{d_{1}}(M_{k}),\ldots,P_{\lambda_{s,k}}^{d_{s}}(M_{k})))\\\
&=&\nu(P_{\lambda_{1,k},\ldots,\lambda_{s,k}}^{d_{1},\ldots,d_{s}}(M_{k})),\end{array}$
(7)

where the last identity is a consequence of Lemma 4.

Now (7), together with (6) and Lemma 5 imply that

$\begin{array}[]{ccl}\displaystyle\sum_{i=1}^{d}m_{i}+d(n-{\mbox{\rm
rank\,}}L)&=&\nu(P_{\lambda_{1,k},\ldots,\lambda_{s,k}}^{d_{1},\ldots,d_{s}}(M_{k}))\\\
&\leq&\displaystyle\nu(P_{\mu}^{d}(M))=\sum_{i=1}^{d}W_{i}(\mu,M)+d(n-{\mbox{\rm
rank\,}}M),\end{array}$

or, equivalently,

$\sum_{i=1}^{d}m_{i}\leq\sum_{i=1}^{d}W_{i}(\mu,M)+d({\mbox{\rm
rank\,}}L-{\mbox{\rm rank\,}}M),$

as wanted.

Now, let us assume that $\mu=\infty$. It is straightforward to see (but we
refer otherwise to [27, Remark 4.3]) that, for a given pencil $L(\lambda)$ and
$\mu\in\overline{\mathbb{C}}$, the identity $W(\mu,L)=W(\mu^{-1},{\rm rev}L)$
holds. It is also immediate to see that ${\mbox{\rm rank\,}}({\rm
rev}L)={\mbox{\rm rank\,}}L$. Moreover, if
$\\{M_{k}(\lambda)\\}_{k\in\mathbb{Z}}$ is a sequence of pencils converging to
$M(\lambda)$, then $\\{{\rm rev}M_{k}\\}_{k\in\mathbb{Z}}$ converges to ${\rm
rev}M$. So let $L(\lambda),M(\lambda)$, and $M_{k}(\lambda)$, as well as
$\lambda_{i,k}$, be as in the statement. Then $\lambda_{i,k}^{-1}$ are
eigenvalues of ${\rm rev}M_{k}$, and they converge to the eigenvalue $0$ of
${\rm rev}M$. By the case just proved for $\mu\in{\mathbb{C}}$, we conclude
that

$\bigcup_{i=1}^{s}W(\lambda_{i,k}^{-1},{\rm rev}M_{k})\prec W(0,{\rm
rev}M)+(h,h,\ldots).$

But, since $W(\lambda_{i,k}^{-1},{\rm rev}M_{k})=W(\lambda_{i,k},M_{k})$ and
$W(0,{\rm rev}M)=W(\infty,M)$, this implies

$\bigcup_{i=1}^{s}W(\lambda_{i,k},M_{k})\prec W(\infty,M)+(h,h,\ldots),$

as claimed. ∎

The following result provides a theoretical answer to the question of deciding
whether a given matrix pencil belongs to the closure of the bundle of another
pencil or not.

######  Theorem 9.

Let $L(\lambda)$ and $M(\lambda)$ be two complex matrix pencils of the same
size. Then $M(\lambda)\in\overline{\cal B}(L)$ if and only if
$M(\lambda)\in\overline{\cal O}(\psi_{c}(L))$, for some map
$\psi:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$.

###### Proof.

Let us first prove the “if” part of the statement. Assume that
$M(\lambda)\in\overline{\cal B}(L)$. Then, there is a sequence of pencils
$\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ such that $M_{k}(\lambda)\in{\cal
B}(L)$ and $\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ converges to $M(\lambda)$.
Since $M_{k}(\lambda)\in{\cal B}(L)$, we conclude:

  * •

$r(M_{k})=r(L)$ and $\ell(M_{k})=\ell(L)$, for all $k\in\mathbb{N}$.

  * •

All $M_{k}(\lambda)$ have the same number of distinct eigenvalues, say $s$,
which is the number of distinct eigenvalues of $L(\lambda)$. Moreover, if
$\lambda_{1},\ldots,\lambda_{s}$ and $\lambda_{1,k},\ldots,\lambda_{s,k}$
denote, respectively, the distinct eigenvalues of $L(\lambda)$ and
$M_{k}(\lambda)$, then $W(\lambda_{i,k},M_{k})=W(\lambda_{i},L)$, for all
$k\in\mathbb{N}$ and all $i=1,\ldots,s$.

If we set $h:={\mbox{\rm rank\,}}L-{\mbox{\rm rank\,}}M$ and
$\sigma_{\varepsilon}(M)=\bigcup_{\mu\in\Lambda(M)}B(\mu,\varepsilon)$ then,
for all $k$ large enough, the pencils $M_{k}(\lambda)$ satisfy (see, for
instance, Lemma 1.1 and its proof in [4]):

  * (i)

$r(M)\prec r(M_{k})+(h,h,\ldots)=r(L)+(h,h,\ldots)$,

  * (ii)

$\ell(M)\prec\ell(M_{k})+(h,h,\ldots)=\ell(L)+(h,h,\ldots)$,

  * (iii)

$W(\mu,M_{k})\prec W(\mu,M)+(h,h,\ldots)$, for all
$\mu\in(\overline{\mathbb{C}}-\sigma_{\varepsilon}(M))\cup\Lambda(M)$.

Now, we are going to see that, by taking a subsequence of
$\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ if necessary, we may assume that, for
any $i=1,\ldots,s$ one of the following conditions holds:

  1. (C1)

$\\{\lambda_{i,k}\\}$ converges to an eigenvalue of $M$, or

  2. (C2)

there is some $\varepsilon>0$ such that
$\lambda_{i,k}\in\overline{\mathbb{C}}-\sigma_{\varepsilon}(M)$, for all $k$
large enough.

Assume that, for some $i=1,\ldots,s$, condition (C2) does not hold. Then, for
any $\varepsilon>0$ there is an infinite subsequence of $\\{\lambda_{i,k}\\}$
included in $\sigma_{\varepsilon}(M)$. Since the spectrum of $M$ is finite,
there is a subsequence, $\\{\lambda_{i,k_{j}}\\}$ that converges to some
eigenvalue of $M(\lambda)$, so $\\{\lambda_{i,k_{j}}\\}$ satisfies (C1). Since
$s$ is a finite number, we can keep going with this procedure, by taking a
finer subsequence if necessary, until we end up with a subsequence of
$\\{M_{k}(\lambda)\\}$, whose eigenvalues $\\{\lambda_{i,k}\\}$ satisfy either
(C1) or (C2), for all $i=1,\ldots,s$.

Now, let us assume that the eigenvalues $\lambda_{i,k}$ converge to some
distinct $\mu_{1},\ldots,\mu_{d}$, for all $i=1,\ldots,t$ and $t\leq s$, where
$\mu_{1},\ldots,\mu_{d}$ are eigenvalues of $M(\lambda)$. Note, however, that
not all eigenvalues of $M_{k}(\lambda)$ must converge to eigenvalues of
$M(\lambda)$, since some of them can satisfy condition (C2) above, and that
not every eigenvalue of $M(\lambda)$ is necessarily obtained in this way,
because some new eigenvalues not coming from eigenvalues of $M_{k}(\lambda)$
could arise in the limit $M(\lambda)$. In particular, we are assuming that the
eigenvalues $\lambda_{i,k}$, for $i=t+1,\ldots,s$, satisfy (C2) above. Let us
gather all eigenvalues $\lambda_{i,k}$ that converge to the same eigenvalue
$\mu_{j}$ of $M(\lambda)$ in the following way: we decompose the set of
indices $\\{1,\ldots,t\\}$ as the union of $d$ disjoint sets of indices,
denoted by $I_{1},\ldots,I_{d}$ in such a way that the eigenvalues
$\lambda_{i,k}$ with $i\in I_{j}$ converge to $\mu_{j}$, for $j=1,\ldots,d$.
This means, in particular, that $M(\lambda)$ has, at least, $d$ different
eigenvalues, namely $\mu_{1},\ldots,\mu_{d}$.

Let $\widetilde{L}:=\psi_{c}(L)$, where
$\psi:\overline{{\mathbb{C}}}\rightarrow\overline{{\mathbb{C}}}$ is such that
$\psi(\lambda_{i})=\mu_{j}$, for $i\in I_{j}$ and $j=1,\ldots,d$,
$\psi(\lambda_{i})=\lambda_{i}$, for $i=t+1,\ldots,s$, and
$\psi(\mu)\not\in\\{\mu_{1},\ldots,\mu_{d},\lambda_{t+1},\ldots,\lambda_{s}\\}$,
for $\mu\not\in\\{\lambda_{1},\ldots,\lambda_{s}\\}$. Then,
$\mu_{1},\ldots,\mu_{d},\lambda_{t+1},\ldots,\lambda_{s}$ are the distinct
eigenvalues of $\widetilde{L}$, and:

  * (iv)

$r(M_{k})=r(L)=r(\widetilde{L})$,

  * (v)

$\ell(M_{k})=\ell(L)=\ell(\widetilde{L})$,

  * (vi)

$W(\mu,\widetilde{L})\prec W(\mu,M)+(h,h,\ldots)$, for all
$\mu\in\overline{\mathbb{C}}$.

Claim (vi) is a consequence of the following facts:

  * •

If $\mu$ is not an eigenvalue of $\widetilde{L}(\lambda)$, then
$(0)=W(\mu,\widetilde{L})\prec W(\mu,M)+(h,h,\ldots)$, since $h\geq 0$.

  * •

$W(\mu_{j},\widetilde{L})=\bigcup_{i\in I_{j}}W(\lambda_{i},L)=\bigcup_{i\in
I_{j}}W(\lambda_{i,k},M_{k})\prec W(\mu_{j},M)+(h,h,\ldots)$, for
$j=1,\ldots,d$, where the first identity is a consequence of the definition of
$\psi_{c}(L)$, and the majorization is a consequence of Theorem 8.

  * •

For the remaining eigenvalues of $\widetilde{L}$, namely $\lambda_{i}$ with
$i=t+1,\ldots,s$, we have
$W(\lambda_{i},\widetilde{L})=W(\lambda_{i},L)=W(\lambda_{i,k},M_{k})\prec
W(\lambda_{i,k},M)+(h,h,\ldots)\break=(0)+(h,h,\ldots)\prec
W(\lambda_{i},M)+(h,h,\ldots)$, where the first majorization is a consequence
of (iii) (which applies because $\lambda_{i,k}$ satisfies (C2)), the
subsequent equality is a consequence of (C2), which implies that
$\lambda_{i,k}$ is not an eigenvalue of $M(\lambda)$, and the last
majorization is immediate.

Now, (i), (ii) together with (iv)–(vi) imply:

  * (a)

$r(M)\prec r(\widetilde{L})+(h,h,\ldots)$,

  * (b)

$\ell(M)\prec\ell(\widetilde{L})+(h,h,\ldots)$,

  * (c)

$W(\mu,\widetilde{L})\prec W(\mu,M)+(h,h,\ldots)$, for all
$\mu\in\overline{\mathbb{C}}$.

But (a)–(c) in turn imply that $M(\lambda)\in\overline{\cal
O}(\widetilde{L})$, according to Theorem 7, and this proves the “if” part of
the statement.

Now, let us prove the “only if” part, so let $L(\lambda)$ be given. We first
prove that $\overline{\cal B}(L)$ contains $\overline{\cal O}(\psi_{c}(L))$,
for all maps $\psi:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$.

We are going to first prove that $\psi_{c}(L)\in\overline{\cal B}(L)$. To this
end, note that ${\cal B}(L)={\cal B}(P\cdot L(\lambda)\cdot Q)$, for any
invertible matrices $P,Q$ of appropriate size. Therefore, we can assume that
$L(\lambda)$ is given in KCF so, without loss of generality, $L(\lambda)$ is
of the form

$L(\lambda)={\mbox{\rm
diag}}(J_{\lambda_{1}}(\lambda),\ldots,J_{\lambda_{s}}(\lambda),\widehat{L}(\lambda)),$

where

$J_{\lambda_{i}}(\lambda):=\displaystyle\bigoplus_{j=1}^{g_{i}}\left(\lambda
I_{\alpha_{j,i}}+J_{\alpha_{j,i}}(\lambda_{i})\right),\qquad\mbox{for
$i=1,\ldots,s$,}$

with $\alpha_{1,i}\geq\cdots\geq\alpha_{g_{i},i}$, for $i=1,\ldots,s$, and
$\Lambda(\widehat{L})\cap\\{\lambda_{1},\ldots,\lambda_{s}\\}=\emptyset$.
Unlike what happens in the first part of the proof (the “if” part), here
$\lambda_{1},\ldots,\lambda_{s}$ denote some distinct eigenvalues, but not
necessarily all the eigenvalues of $L(\lambda)$. Also, for simplicity we are
assuming that $\lambda_{1},\ldots,\lambda_{s}$ are all finite eigenvalues. For
the case where some $\lambda_{k}$, for $1\leq k\leq s$, is the infinite
eigenvalue, the arguments are also valid just replacing $\lambda
I_{\alpha_{j,k}}+J_{\alpha_{j,k}}(\lambda_{k})$ by $I_{\alpha_{j,k}}+\lambda
N_{\alpha_{j,k}}$, and incorporating the appropriate changes in the
corresponding blocks of the forthcoming arguments.

Let us first prove that, if $\psi\in\Psi$ is such that
$\psi^{-1}(\mu)=\\{\lambda_{1},\ldots,\lambda_{s}\\}\cup S$, with
$S\cap\Lambda(L)=\emptyset$, for some $\mu\in\overline{{\mathbb{C}}}$, then
$\psi_{c}(L)\in\overline{\cal B}(L)$.

By definition of $\psi_{c}(L)$, it suffices to find some
$\widetilde{L}(\lambda)$ such that $\widetilde{L}(\lambda)\in\overline{\cal
B}(L)$ and ${\rm KCF}(\widetilde{L})={\mbox{\rm
diag}}(\widetilde{J}_{\mu}(\lambda),\widehat{L}(\lambda))$, where $\mu$ is the
only eigenvalue of $\widetilde{J}_{\mu}(\lambda)$ (and is not an eigenvalue of
$\widehat{L}(\lambda)$), and such that
$W(\mu,\widetilde{L})=\bigcup_{i=1}^{s}W(\lambda_{i},L)$.

Without loss of generality, we may assume that $g_{1}\geq\cdots\geq g_{s}$.
For brevity, we denote by $E_{m}(i_{1},\ldots,i_{t})$ the $m\times m$ matrix
having an entry equal to $1$ in the positions
$(i_{1},i_{1}+1),\ldots,(i_{t},i_{t}+1)$, and zeroes elsewhere. Then, set

$\begin{array}[]{rcl}L_{k}(\lambda)&:=&{\mbox{\rm
diag}}\left(\displaystyle\bigoplus_{j=1}^{g_{s}}\left(\bigoplus_{i=1}^{s}\left(\lambda
I_{\alpha_{j,i}}+J_{\alpha_{j,i}}\left(\mu+\frac{i}{k}\right)\right)\right.+\right.\\\
&&\left.E_{\alpha_{j,1}+\cdots+\alpha_{j,s}}(\alpha_{j,1},\alpha_{j,1}+\alpha_{j,2},\ldots,\alpha_{j,1}+\cdots+\alpha_{j,s-1})\right),\\\
&&\displaystyle\bigoplus_{j=g_{s}+1}^{g_{s-1}}\left(\bigoplus_{i=1}^{s-1}\left(\lambda
I_{\alpha_{j,i}}+J_{\alpha_{j,i}}\left(\mu+\frac{i}{k}\right)\right)\right.+\\\
&&\left.E_{\alpha_{j,1}+\cdots+\alpha_{j,s-1}}(\alpha_{j,1},\alpha_{j,1}+\alpha_{j,2},\ldots,\alpha_{j,1}+\cdots+\alpha_{j,s-2})\right),\\\
&&\left.,\ldots,\displaystyle\bigoplus_{j=g_{2}+1}^{g_{1}}\left(\lambda
I_{\alpha_{j,1}}+J_{\alpha_{j,1}}\left(\mu+\frac{1}{k}\right)\right),\widehat{L}(\lambda)\right),\end{array}$

where the addends of the form $\bigoplus_{j=g_{\ell}+1}^{g_{\ell-1}}$, with
$g_{\ell-1}=g_{\ell}$ are empty. Note that:

  * •

$L_{k}(\lambda)\in{\cal B}(L)$, since, following the same notation as above,

${\rm KCF}(L_{k})={\mbox{\rm
diag}}(J_{\mu+\frac{1}{k}}(\lambda),\ldots,J_{\mu+\frac{s}{k}}(\lambda),\widehat{L}(\lambda)),$

that is, $J_{\mu+\frac{i}{k}}(\lambda)$ is obtained from
$J_{\lambda_{i}}(\lambda)$ just replacing the eigenvalue $\lambda_{i}$ by the
eigenvalue $\mu+\frac{i}{k}$, for $i=1,\ldots,s$.

  * •

The sequence $\\{L_{k}\\}_{k\in\mathbb{N}}$ converges to

$\begin{array}[]{ccl}\widetilde{L}(\lambda)&:=&{\mbox{\rm
diag}}\left(\displaystyle\bigoplus_{j=1}^{g_{s}}\left(\bigoplus_{i=1}^{s}\left(\lambda
I_{\alpha_{j,i}}+J_{\alpha_{j,i}}(\mu)\right)\right.+\right.\\\
&&\left.E_{\alpha_{j,1}+\cdots+\alpha_{j,s}}(\alpha_{j,1},\alpha_{j,1}+\alpha_{j,2},\ldots,\alpha_{j,1}+\cdots+\alpha_{j,s-1})\right),\\\
&&\displaystyle\bigoplus_{j=g_{s}+1}^{g_{s-1}}\left(\bigoplus_{i=1}^{s-1}\left(\lambda
I_{\alpha_{j,i}}+J_{\alpha_{j,i}}(\mu)\right)\right.+\\\
&&\left.E_{\alpha_{j,1}+\cdots+\alpha_{j,s-1}}(\alpha_{j,1},\alpha_{j,1}+\alpha_{j,2},\ldots,\alpha_{j,1}+\cdots+\alpha_{j,s-2})\right),\\\
&&\left.,\ldots,\displaystyle\bigoplus_{j=g_{2}+1}^{g_{1}}\left(\lambda
I_{\alpha_{j,1}}+J_{\alpha_{j,1}}(\mu)\right),\widehat{L}(\lambda)\right)\\\
&=&{\mbox{\rm
diag}}(\widetilde{J}_{\mu}(\lambda),\widehat{L}(\lambda)),\end{array}$

where $\mu$ is the only eigenvalue of $\widetilde{J}_{\mu}(\lambda)$, and
$W(\mu,\widetilde{J}_{\mu})=\bigcup_{i=1}^{s}W(\lambda_{i},J_{\lambda_{i}})=\bigcup_{i=1}^{s}W(\lambda_{i},L)$,
by Lemma 6 (note that, by
construction,$S(\mu,\widetilde{J}_{\mu})=\Sigma_{i=1}^{s}S(\lambda_{i},L)$).

Therefore, $\widetilde{L}(\lambda)\in\overline{\cal B}(L)$ and, moreover, it
is obtained from $L(\lambda)$ after coalescing
$\lambda_{1},\ldots,\lambda_{s}$ to the eigenvalue $\mu$, and keeping the
remaining eigenvalues of $L(\lambda)$ unchanged.

Using the same approach as above, applied to the block $\widehat{L}(\lambda)$,
for the other values $\widehat{\mu}\in\overline{\mathbb{C}}$ such that
$\psi^{-1}(\widehat{\mu})\cap\Lambda(\widehat{L})\neq\emptyset$, we can
construct a matrix pencil $L^{\prime}(\lambda)\in\overline{\cal B}(L)$ such
that $L^{\prime}(\lambda)=\psi_{c}(L)$ and
$\psi^{-1}(\mu)=\\{\lambda_{1},\ldots,\lambda_{s}\\}\cup S$, with
$S\cap\Lambda(L)=\emptyset$. Since $\\{\lambda_{1},\ldots,\lambda_{s}\\}$ is
any subset of $\Lambda(L)$, this proves that $\psi_{c}(L)\in\overline{\cal
B}(L)$ for any $\psi\in\Psi$, as wanted.

Since the particular values of $\mu$ and $\widehat{\mu}$ are not relevant, we
conclude that ${\cal B}(\psi_{c}(L))\subseteq\overline{\cal B}(L)$, for any
$\psi\in\Psi$. This immediately implies $\overline{\cal
B}(\psi_{c}(L))\subseteq\overline{\cal B}(L)$, by definition of closure.

Finally, if $M(\lambda)\in\overline{\cal O}(\psi_{c}(L))$, for some
$\psi\in\Psi$, then $M(\lambda)\in\overline{\cal
O}(\psi_{c}(L))\subseteq\overline{\cal B}(\psi_{c}(L))\subseteq\overline{\cal
B}(L)$, and we are done. ∎

######  Remark 10.

Note that Theorem 9 is equivalent to the identity:

$\overline{\cal B}(L)=\displaystyle\bigcup_{\psi\in\Psi}\overline{{\cal
O}}(\psi_{c}(L)),$ (8)

for any complex matrix pencil $L(\lambda)$.

There is a striking difference between the statement of Theorem 7 and the one
of Theorem 9. More precisely, in Theorem 7 it is implicitly assumed that
$P_{2}(\lambda)\in\overline{\cal O}(P_{1})$ implies $\overline{\cal
O}(P_{2})\subseteq\overline{\cal O}(P_{1})$. This is clearly true, since
$P_{2}(\lambda)\in\overline{\cal O}(P_{1})$ implies that
$RP_{2}(\lambda)S\in\overline{\cal O}(P_{1})$, for any pair of invertible
matrices $R$ and $S$, which in turns implies ${\cal
O}(P_{2})\subseteq\overline{\cal O}(P_{1})$ and this, by definition of
closure, implies $\overline{\cal O}(P_{2})\subseteq\overline{\cal O}(P_{1})$.
However, Theorem 9 does not yet provide a characterization for the inclusion
of bundle closures, since it is not so clear that $M(\lambda)\in\overline{\cal
B}(L)$ implies $\overline{{\cal B}}(M)\subseteq\overline{{\cal B}}(L)$. This
is, indeed, true, and it is a consequence of Lemma 11.

######  Lemma 11.

Let $L(\lambda)$ and $M(\lambda)$ be two complex matrix pencils of the same
size. If $M(\lambda)\in\overline{\cal B}(L)$ then $\overline{\cal
B}(M)\subseteq\overline{\cal B}(L)$.

###### Proof.

We are first going to prove that, if $L(\lambda)$ and $M(\lambda)$ are as in
the statement and $M^{\prime}(\lambda)\in{\cal B}(M)$, then
$M^{\prime}(\lambda)\in\overline{\cal B}(L)$.

Since $M(\lambda)\in\overline{\cal B}(L)$, Theorem 9 implies that
$M(\lambda)\in\overline{{\cal O}}(\psi_{c}(L))$, for some $\psi\in\Psi$. Also,
since $M^{\prime}(\lambda)\in{\cal B}(M)$, then
$M^{\prime}(\lambda)=R\,\varphi(K_{M})\,S$, for some $R,S$ invertible and
$\varphi\in\Phi$ (namely, $\varphi$ is one-to-one). In particular,
$\mu\in\Lambda(M)$ if and only if $\varphi(\mu)\in\Lambda(M^{\prime})$ or,
since $\varphi$ is one-to-one, $\varphi^{-1}(\mu)\in\Lambda(M)$ if and only if
$\mu\in\Lambda(M^{\prime})$, and, moreover:

$W(\varphi^{-1}(\mu),M)=W(\mu,M^{\prime}).$ (9)

Now, we claim that

$M^{\prime}(\lambda)\in\overline{{\cal O}}((\varphi\circ\psi)_{c}(L)).$ (10)

Note that this immediately implies that $M^{\prime}(\lambda)\in\overline{\cal
B}(L)$, by Theorem 9 again.

In order to prove (10), we use Theorem 7. More precisely, we need to prove,
for $h:={\mbox{\rm rank\,}}(\varphi\circ\psi)_{c}(L)-{\mbox{\rm
rank\,}}M^{\prime}$:

  1. (a’)

$r(M^{\prime})\prec r((\varphi\circ\psi)_{c}(L))+(h,h,\ldots)$.

  2. (b’)

$\ell(M^{\prime})\prec\ell((\varphi\circ\psi)_{c}(L))+(h,h,\ldots)$.

  3. (c’)

$W(\mu,(\varphi\circ\psi)_{c}(L))\prec W(\mu,M^{\prime})+(h,h,\ldots)$, for
all $\mu\in\overline{\mathbb{C}}$.

Using $M^{\prime}(\lambda)\in{\cal B}(M)$ and Definition 1, we get

$\displaystyle r(M)$ $\displaystyle=$ $\displaystyle r(M^{\prime}),$ (11)
$\displaystyle\ell(M)$ $\displaystyle=$ $\displaystyle\ell(M^{\prime}),$ (12)
$\displaystyle r(\psi_{c}(L))$ $\displaystyle=$ $\displaystyle
r((\varphi\circ\psi)_{c}(L))=r(L),$ (13) $\displaystyle\ell(\psi_{c}(L))$
$\displaystyle=$ $\displaystyle\ell((\varphi\circ\psi)_{c}(L))=\ell(L).$ (14)

In particular, ${\mbox{\rm rank\,}}M={\mbox{\rm rank\,}}M^{\prime}$ and
${\mbox{\rm rank\,}}(\varphi\circ\psi)_{c}(L)={\mbox{\rm rank\,}}L={\mbox{\rm
rank\,}}\psi_{c}(L)$. This implies that

$h={\mbox{\rm rank\,}}\psi_{c}(L)-{\mbox{\rm rank\,}}M={\mbox{\rm
rank\,}}L-{\mbox{\rm rank\,}}M.$ (15)

Now, to see that claims (a’) and (b’) are true note that, since
$M(\lambda)\in\overline{{\cal O}}(\psi_{c}(L))$, by Theorem 7, we have:

  1. (a)

$r(M)\prec r(\psi_{c}(L))+(h,h,\ldots)$.

  2. (b)

$\ell(M)\prec\ell(\psi_{c}(L))+(h,h,\ldots)$.

Then, (a) and (b) together with (11)–(15) immediately imply (a’) and (b’).

To prove (c’), let $\mu\in\overline{\mathbb{C}}$. Then,

$\begin{array}[]{ccl}W(\mu,(\varphi\circ\psi)_{c}(L))&=&\bigcup_{\mu_{i}\in(\varphi\circ\psi)^{-1}(\mu)}W(\mu_{i},L)=\bigcup_{\mu_{i}\in(\psi)^{-1}(\varphi^{-1}(\mu))}W(\mu_{i},L)\\\
&=&W(\varphi^{-1}(\mu),\psi_{c}(L))\prec
W(\varphi^{-1}(\mu),M)+(h,h,\ldots)\\\
&=&W(\mu,M^{\prime})+(h,h,\ldots),\end{array}$

where the majorization is a consequence of $M(\lambda)\in\overline{{\cal
O}}(\psi_{c}(L))$, and the last identity is (9). Note that, by (15), the
values of $h$ in the last two expressions correspond to ${\mbox{\rm
rank\,}}\psi_{c}(L)-{\mbox{\rm rank\,}}M$ in the first one, and to ${\mbox{\rm
rank\,}}(\varphi\circ\psi)_{c}(L)-{\mbox{\rm rank\,}}M^{\prime}$ in the second
one.

We have seen so far that, if $M(\lambda)\in\overline{\cal B}(L)$, then ${\cal
B}(M)\subseteq\overline{\cal B}(L)$. By definition of closure, this implies
$\overline{\cal B}(M)\subseteq\overline{\cal B}(L)$, as wanted. ∎

Theorem 9, together with Lemma 11, immediately imply Theorem 12.

######  Theorem 12.

(Characterization of the inclusion between bundle closures). Let $L(\lambda)$
and $M(\lambda)$ be two matrix pencils of the same size. Then $\overline{\cal
B}(M)\subseteq\overline{\cal B}(L)$ if and only if
$M(\lambda)\in\overline{\cal O}(\psi_{c}(L))$, for some map
$\psi:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$.

In other words, Theorem 12 says that $\overline{\cal
B}(M)\subseteq\overline{\cal B}(L)$ if and only if ${\rm KCF}(M)$ is obtained
from ${\rm KCF}(L)$ after coalescing eigenvalues and applying the dominance
rules in Theorem 7. The same result is stated in [18, Th. 2.6] for bundles of
matrices instead of matrix pencils (see Theorem 20).

##  4 Bundles are open in their closure

The fact that bundles are open in their closure will be an almost immediate
consequence of the fact that $\overline{{\cal B}}(L)$ is the union of ${\cal
B}(L)$ with a finite number of other bundles whose closure is strictly
included in $\overline{{\cal B}}(L)$. This fact is implicitly assumed in the
developments carried out in [18]. More precisely, the set of matrix pencils is
claimed to be a stratified manifold (see [18, p. 670]), namely “the union of
non-intersecting manifolds whose closure is the finite union of itself with
strata of smaller dimensions”. The strata can be either orbits (for singular
matrix pencils) or bundles. Moreover, much effort has been paid, by different
authors, to analyze and/or to describe the stratification of bundles (and
orbits) of general and structured sets of matrices and matrix pencils (to cite
just a few of the works dealing with bundles, see [15, 18, 29] and the
references therein), as well as for matrix polynomials [10, 16]. In this
context, a Java-based tool, called Stratigraph, has been developed [19] for
computing and displaying the closure hierarchies of the strata (bundles and
orbits). However, we have not found in the literature a proof of the fact that
the closure of a bundle is indeed the union of itself with other bundles of
smaller dimension (for the case of orbits, it is a consequence of general
results on group actions, see, for instance, the Proposition in [23, p. 60]).

The main goal of this section is to prove that bundles are open in their
closure. We will prove it for bundles of matrix pencils under strict
equivalence (in Theorem 17), as well as for bundles of matrices under
similarity (in Section 4.1) and for bundles of matrix polynomials (in Section
4.2).

For bundles of matrix pencils, and as mentioned above, we are going to prove
first a more general result (Theorem 16), namely that $\overline{{\cal B}}(L)$
is the union of ${\cal B}(L)$ together with a finite number of other bundles
whose closure in strictly included in $\overline{{\cal B}}(L)$.

We start with the following well-known result, that we include here for the
sake of completeness.

######  Lemma 13.

The number of different bundles of complex matrix pencils with size $m\times
n$ is finite.

###### Proof.

Each bundle of a fixed size $m\times n$ is determined by:

  * •

The number of different eigenvalues, which is less than or equal to
$\min\\{m,n\\}$.

  * •

The sizes of the Jordan blocks associated with each eigenvalue (that is, the
partial multiplicities), which are bounded by $\min\\{m,n\\}$.

  * •

The minimal indices (left and right), whose sums are bounded by $m$ (left) and
$n$ (right).

Since there is only a finite number of all these three quantities, for a fixed
size $m\times n$, the number of different $m\times n$ bundles is finite. ∎

######  Remark 14.

Lemma 13 is in contrast with what happens with orbits, since the number of
different orbits of $m\times n$ matrix pencils is infinite (since there are
infinitely many different eigenvalues).

The dimension of ${\cal O}(L)$, denoted $\dim{\cal O}(L)$, is the dimension of
${\cal O}(L)$ when considered as a differentiable manifold (see, for instance,
[5]).

The following result provides a description of the closure of a bundle as the
union of orbits (and not of orbit closures, as in (8)), and is a first step to
prove Theorem 16, where the closure of a bundle is described as a union of
bundles.

######  Theorem 15.

Let $L(\lambda)$ be an $m\times n$ complex matrix pencil. Then $\overline{\cal
B}(L)={\cal B}(L)\cup\widetilde{\cal B}$, where $\widetilde{\cal B}$ is a
union of orbits. Each of these orbits, say ${\cal O}(M)$, satisfies one of the
following conditions:

  * •

$\dim{\cal O}(M)<\dim{\cal O}(L)$, or

  * •

${\cal O}(M)={\cal O}(\psi_{c}(L))$, for some $\psi\in\Psi$ which is not one-
to-one on $\Lambda(L)$.

###### Proof.

Let us first prove the following identity:

$\dim{\cal O}(\psi_{c}(L))=\dim{\cal O}(L),\quad\mbox{for all $\psi\in\Psi$}.$
(16)

If $\psi\in\Phi$, then the identity in (16) is an immediate consequence of the
codimension count in [5]. More precisely, the codimension (and, as a
consequence, the dimension) of the orbit of a pencil $L(\lambda)$ depends on
the sizes of the regular and singular blocks in KCF($L$), in such a way that
for two pencils in the same bundle the (co)dimensions of their orbits are the
same (see the quantities 1–5 in the codimension count of [5, p. 65]). Also all
quantities except the one numbered by 1 (and so-called “codimension of the
Jordan structure”) in [5, p. 65] are the same for $L(\lambda)$ and
$\psi_{c}(L)$, even when $\psi$ is not one-to-one, because these two pencils
have the same singular blocks, and the size of their regular part (the “Jordan
structure”) is the same as well. A closer look to quantity 1 allows us to
conclude that it is also the same for $L(\lambda)$ and $\psi_{c}(L)$ when
$\psi$ is not one-to-one. To see this, let us first recall that this quantity
is equal to

$c_{Jor}(L)=\sum_{\mu\in\overline{{\mathbb{C}}}}\left(S_{1}(\mu,L)+3S_{2}(\mu,L)+5S_{3}(\mu,L)+\cdots\right),$

where $S(\mu,L)=(S_{1}(\mu,L),S_{2}(\mu,L),S_{3}(\mu,L),\ldots)$ is the Segre
characteristic of $\mu$ in $L(\lambda)$, namely, the list of sizes of the
Jordan blocks in KCF($L$) associated with $\mu$, ordered in non-increasing
order. If $\psi^{-1}(\mu)=\\{\mu_{1},\ldots,\mu_{k}\\}\cup S$, with
$S\cap\Lambda(L)=\emptyset$, then, by Definition 1,
$W(\mu,\psi_{c}(L))=\bigcup_{i=1}^{k}W(\mu_{i},L)$, which is equivalent to
$S(\mu,\psi_{c}(L))=\sum_{i=1}^{k}S(\mu_{i},L)$, by Lemma 6. Then

$\begin{array}[]{ccl}c_{Jor}(\psi_{c}(L))&=&\sum_{\mu\in\overline{{\mathbb{C}}}}\left(S_{1}(\mu,\psi_{c}(L))+3S_{2}(\mu,\psi_{c}(L))+5S_{3}(\mu,\psi_{c}(L))+\cdots\right)\\\
&=&\sum_{\mu\in\overline{{\mathbb{C}}}}\left(S_{1}(\mu,L)+3S_{2}(\mu,L)+5S_{3}(\mu,L)+\cdots\right)=c_{Jor}(L).\end{array}$

Now, let us decompose the identity in (8) as follows:

$\overline{\cal B}(L)=\left(\bigcup_{\varphi\in\Phi}\overline{\cal
O}(\varphi(L))\right)\cup\left(\bigcup_{\psi\in\Psi_{\Lambda(L)}}\overline{\cal
O}(\psi_{c}(L))\right),$ (17)

where $\Psi_{\Lambda(L)}$ is the set of mappings from
$\overline{{\mathbb{C}}}$ to itself which are not one-to-one on $\Lambda(L)$.
In order to obtain (17) from (8), just note that $\varphi_{c}(L)=\varphi(L)$
when $\varphi\in\Phi$, so the decomposition in (17) comes from splitting the
union in (8) into two pieces: one corresponding to the maps from
$\overline{\mathbb{C}}$ to $\overline{{\mathbb{C}}}$ which are one-to-one on
$\Lambda(L)$, and the other one corresponding to the remaining maps. Now,
since the closure of an orbit is the union of the orbit itself and other
orbits with smaller dimension (see the Proposition in [23, p. 60]), we can
write

$\overline{\cal B}(L)={\cal B}(L)\cup\widetilde{\cal B}_{1}\cup\widetilde{\cal
B}_{2}\cup\widetilde{\cal B}_{3},$

where

${\cal B}(L)=\bigcup_{\varphi\in\Phi}{\cal O}(\varphi(L)),\quad\widetilde{\cal
B}_{1}=\bigcup_{\psi\in\Psi_{\Lambda(L)}}{\cal O}(\psi_{c}(L)),$

the set $\widetilde{\cal B}_{2}$ is a union of orbits with dimension smaller
than ${\cal O}(\varphi(L))$, for any $\varphi\in\Phi$ (coming from the left
union in the right-hand side of (17)), and $\widetilde{\cal B}_{3}$ is a union
of orbits with dimension smaller than ${\cal O}(\psi_{c}(L))$, for any
$\psi\in\Psi_{\Lambda(L)}$ (coming from the right union in the right-hand side
of (17)).

By (16), the orbits in $\widetilde{B}_{2}$ have dimension strictly smaller
than the dimension of ${\cal O}(L)$. Similarly, $\widetilde{\cal B}_{1}$ is a
union of orbits of the form ${\cal O}(\psi_{c}(L))$, they all having the same
dimension as ${\cal O}(L)$, and $\widetilde{\cal B}_{3}$ is a union of orbits
with dimension strictly smaller than the dimension of ${\cal O}(L)$. This
concludes the proof. ∎

The following result provides a description of the closure of the bundle of a
pencil as a finite union of bundles. This is the counterpart of the
corresponding result for orbits, namely that the closure of an orbit is a
union of orbits, which is already known (and has been used in the proof of
Theorem 15). Nevertheless, we emphasize the important difference that the
closure of the orbit of a pencil may be the union of infinitely many orbits.

######  Theorem 16.

Let $L(\lambda)$ be an $m\times n$ pencil. Then, there is a finite number of
different $m\times n$ pencils, $L_{1}(\lambda),\ldots,L_{\ell}(\lambda)$
(with, say, $L_{1}=L$), satisfying ${\cal B}(L_{i})\neq{\cal B}(L_{j})$, for
$i\neq j$, and such that

$\overline{\cal B}(L)=\bigcup_{i=1}^{\ell}{\cal B}(L_{i}).$ (18)

Moreover, $\overline{\cal B}(L_{i})$ is strictly included in $\overline{\cal
B}(L)$, for $i\neq 1$.

###### Proof.

Since, by Lemma 13, the number of different bundles of $m\times n$ matrix
pencils is finite, let ${\cal B}(L_{1}),\ldots,{\cal B}(L_{d})$ be these
bundles, for some $m\times n$ matrix pencils
$L_{1}(\lambda),\ldots,L_{d}(\lambda)$. Without loss of generality (after a
reordering, if necessary), we may assume that
$L_{1}(\lambda),\ldots,L_{\ell}(\lambda)\in\overline{\cal B}(L)$, and
$L_{\ell+1}(\lambda),\ldots,L_{d}(\lambda)\not\in\overline{\cal B}(L)$, for
some $0\leq\ell\leq d$. By Lemma 11, $\bigcup_{i=1}^{\ell}{\cal
B}(L_{i})\subseteq\overline{\cal B}(L)$. The reverse inclusion is immediate,
since $\widetilde{L}(\lambda)\in\overline{\cal B}(L)$ implies that ${\cal
B}(\widetilde{L})\subseteq\overline{\cal B}(L)$, again by Lemma 11, so ${\cal
B}(\widetilde{L})={\cal B}(L_{i})$, for some $1\leq i\leq\ell$, so
$\widetilde{L}(\lambda)\in\bigcup_{i=1}^{\ell}{\cal B}(L_{i})$.

Finally, let us prove that $\overline{\cal B}(L_{i})$ is strictly included in
$\overline{\cal B}(L)$, for $i\neq 1$. First, note that, by Theorem 15, either

  * (a)

$\dim{\cal O}(L_{i})<\dim{\cal O}(L)$, or

  * (b)

${\cal O}(L_{i})={\cal O}(\psi_{c}(L))$, for some $\psi\in\Psi$ which is not
one-to-one on $\Lambda(L)$.

In case (a) we clearly have

$\dim{\cal O}(L_{i})\leq\dim{\cal O}(L),$ (19)

and (16) implies that (19) also holds in case (b).

Assume, by contradiction, that $\overline{{\cal B}}(L_{i})=\overline{{\cal
B}}(L)$, for some $i\neq 1$. This implies, in particular, that
$L(\lambda)\in\overline{\cal B}(L_{i})$. Thus, Theorem 15 applied to
$\overline{\cal B}(L_{i})$ implies that either

  * (c)

$\dim{\cal O}(L)<\dim{\cal O}(L_{i})$, or

  * (d)

${\cal O}(L)={\cal O}(\widetilde{\psi}_{c}(L_{i}))$, for some
$\widetilde{\psi}\in\Psi$ which is not one-to-one on $\Lambda(L_{i})$.

Condition (c) does not hold because of (19), so (d) must hold. But (d)
combined with (16) implies $\dim{\cal O}(L)=\dim{\cal
O}(\widetilde{\psi}_{c}(L_{i}))=\dim{\cal O}(L_{i})$, which implies that
condition (a) does not hold. So, (b) and (d) hold simultaneously. But (b)
implies that the number of distinct eigenvalues of $L_{i}$ is strictly less
than the number of distinct eigenvalues of $L$, while (d) implies the
opposite, which is a contradiction. Thus, we conclude that $\overline{\cal
B}(L_{i})$ is strictly included in $\overline{\cal B}(L)$. ∎

Now, we are in the position to prove the main result in this section.

######  Theorem 17.

Let $L(\lambda)$ be an $m\times n$ matrix pencil. Then ${\cal B}(L)$ is an
open set in its closure.

###### Proof.

Let $M(\lambda)\in{\cal B}(L)$. We want to prove that there is an open
neighborhood of $M(\lambda)$, say $U$, such that $U\cap\overline{\cal
B}(L)\subseteq{\cal B}(L)$. Let us proceed by contradiction. Then, there is a
sequence, $\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$, of matrix pencils that
converges to $M(\lambda)$, with $M_{k}(\lambda)\in\overline{\cal B}(L)$ but
$M_{k}(\lambda)\not\in{\cal B}(L)$. By Theorem 16, $\overline{\cal
B}(L)=\bigcup_{i=1}^{\ell}{\cal B}(L_{i})$, with $L_{1}(\lambda)=L(\lambda)$.
Since this is a finite union and $M_{k}(\lambda)\not\in{\cal B}(L)$, there is
an infinite number of terms in the sequence
$\\{M_{k}(\lambda)\\}_{k\in\mathbb{N}}$ belonging to some ${\cal B}(L_{i})$,
with $i\neq 1$. Without loss of generality, let us assume that $i=2$ and,
considering a subsequence if necessary, assume that $M_{k}(\lambda)\in{\cal
B}(L_{2})$, for all $k\in\mathbb{N}$. But $M_{k}(\lambda)\rightarrow
M(\lambda)$ implies that $M(\lambda)\in\overline{\cal B}(L_{2})$. Note that
the hypothesis $M(\lambda)\in{\cal B}(L)$ implies ${\cal B}(M)={\cal B}(L)$,
so Lemma 11 in turn implies that $\overline{\cal B}(L)\subseteq\overline{\cal
B}(L_{2})$, which in turn implies $\overline{\cal B}(L)=\overline{\cal
B}(L_{2})$, and this is a contradiction with the last claim in Theorem 16. ∎

######  Remark 18.

We want to highlight that Theorem 7.5 in [29] is false. This theorem states
the following, with the notation of this paper:

$\overline{\cal B}(L)=\bigcup_{\psi\in\Psi}\overline{\cal O}(\psi(L)).$ (20)

Comparing (20) with (8), we see that the difference is that in (20) the
function $\psi$ is used instead of the right one, namely $\psi_{c}(L)$. In
other words, (20) is missing all pencils that are obtained by coalescing
eigenvalues of $L(\lambda)$.

A counterexample of (20) is the following. Let

$L(\lambda):=\left[\begin{array}[]{c|cc}\lambda-3&0&0\\\ \hline\cr
0&\lambda-2&1\\\ 0&0&\lambda-2\end{array}\right]={\mbox{\rm
diag}}(\lambda+J_{1}(3),\lambda I_{2}+J_{2}(2)),$

and

$L_{k}(\lambda):=\begin{bmatrix}\lambda-(2+1/k)&1&0\\\ 0&\lambda-2&1\\\
0&0&\lambda-2\end{bmatrix}.$

Note that

$\widehat{L}(\lambda):=\lim_{k\rightarrow\infty}L_{k}(\lambda)=\begin{bmatrix}\lambda-2&1&0\\\
0&\lambda-2&1\\\ 0&0&\lambda-2\end{bmatrix}=\lambda I_{3}+J_{3}(2).$

Then

$L_{k}(\lambda)\in{\cal
O}\left(\left[\begin{array}[]{c|cc}\lambda-(2+1/k)&0&0\\\ \hline\cr
0&\lambda-2&1\\\ 0&0&\lambda-2\end{array}\right]\right)\subseteq{\cal
B}(L),\quad\mbox{for all $k\in\mathbb{N}$,}$

which implies that

$\widehat{L}(\lambda)\in\overline{\cal B}(L).$ (21)

We are going to prove that

$\widehat{L}(\lambda)\not\in\bigcup_{\psi\in\Psi}\overline{\cal O}(\psi(L)).$
(22)

Note that equations (21) and (22) contradict (20).

In order to see (22), note first that

${\cal O}(\psi(L))={\cal
O}\left(\left[\begin{array}[]{c|cc}\lambda-\psi(3)&0&0\\\ \hline\cr
0&\lambda-\psi(2)&1\\\ 0&0&\lambda-\psi(2)\end{array}\right]\right)={\cal
O}\left(\left[\begin{array}[]{c|cc}\lambda-a&0&0\\\ \hline\cr 0&\lambda-b&1\\\
0&0&\lambda-b\end{array}\right]\right),$

with $a,b\in\overline{\mathbb{C}}$ arbitrary (if $a$ or $b$ is $\infty$ then
the pencil in the right-hand side should be replaced accordingly by either
${\mbox{\rm diag}}(\lambda-a,\lambda N_{2}+I_{2})$, ${\mbox{\rm
diag}}(1,\lambda I_{2}+J_{2}(b)),$ or ${\mbox{\rm diag}}(1,\lambda
N_{2}+I_{2})$).

If $a=b$, then

$\widehat{L}(\lambda)\not\in\overline{\cal
O}(\left(\left[\begin{array}[]{c|cc}\lambda-a&0&0\\\ \hline\cr
0&\lambda-a&1\\\ 0&0&\lambda-a\end{array}\right]\right)$

(with, again, performing the appropriate changes if either $a=b=\infty$). To
see this, note that, in order for $\widehat{L}(\lambda)$ to belong to the
closure of the orbit in the right-hand side of the previous equation, it
should be $a=2$. However, the closure relationships of matrix orbits in
Theorem 7 guarantee that $\widehat{L}$ does not belong to this closure (the
majorization $(2,1,0)\prec(1,1,1)$ does not hold).

If $a\neq b$, then

$\widehat{L}(\lambda)\not\in\overline{\cal
O}(\left(\left[\begin{array}[]{c|cc}\lambda-a&0&0\\\ \hline\cr
0&\lambda-b&1\\\ 0&0&\lambda-b\end{array}\right]\right)$

because the eigenvalues of regular pencils are continuous functions of the
entries of their coefficients, thus $\widehat{L}(\lambda)$, that has only one
eigenvalue equal to $2$ (with multiplicity $3$) cannot be the limit of a
sequence of pencils with two different eigenvalues equal to $a$ and $b$
(fixed).

###  4.1 Bundles of matrices under similarity

We can also prove that bundles of matrices under similarity are open in their
closure using similar arguments to the ones for bundles of matrix pencils
above in this section, combined with results from [18]. More precisely, for a
matrix $A\in{\mathbb{C}}^{n\times n}$, we define

$\begin{array}[]{ccll}{\cal O}(A)&:=&\\{PAP^{-1}\,:\ P\in\mbox{\rm
GL}_{n}({\mathbb{C}})\\}&\mbox{(orbit of $A$),}\\\ {\cal
B}(A)&:=&\displaystyle\bigcup_{\varphi\in\Phi}{\cal
O}(\varphi(A))&\mbox{(bundle of $A$),}\end{array}$

where now $\Phi$ is the set of one-to-one maps from ${\mathbb{C}}$ to itself
(since the infinite eigenvalue does not apply for matrices) and $\varphi(A)$
is any matrix similar to the one obtained from the JCF of $A$ after replacing
the Jordan blocks with eigenvalue $\mu\in\mathbb{C}$ by Jordan blocks of the
same size with eigenvalue $\varphi(\mu)$ for any eigenvalue $\mu$ of $A$.
Similarly, if $\Psi$ denotes the set of maps from ${\mathbb{C}}$ to itself
(not necessarily one-to-one), then we can introduce the following notion of
coalescence for matrices:

######  Definition 19.

(Coalescence of eigenvalues of matrices). Let $A\in{\mathbb{C}}^{n\times n}$
be a matrix with different eigenvalues $\mu_{1},\ldots,\mu_{s}$, and let
$\psi\in\Psi$. Then, $\psi_{c}(A)$ is any $n\times n$ matrix satisfying the
following property:

  * •

$W(\mu,\psi_{c}(A))=\bigcup_{\mu_{i}\in\psi^{-1}(\mu)}W(\mu_{i},A)$, for all
$\mu\in{\mathbb{C}}$.

We say that the eigenvalues $\mu_{i_{1}},\ldots,\mu_{i_{d}}$ of $A$ have
coalesced to the eigenvalue $\mu$ in $\psi_{c}(A)$ if
$\psi^{-1}(\mu)=\\{\mu_{i_{1}},\ldots,\mu_{i_{d}}\\}\cup S$, with
$S\cap\Lambda(A)=\emptyset$.

Observe that all matrices $\psi_{c}(A)$ in Definition 19 are similar to each
other, since they all have the same Jordan canonical form.

With these definitions, the characterization of the bundle closure inclusion
for matrices under similarity, provided in [18, Th. 2.6], can be stated as
follows:

######  Theorem 20.

[18, Th. 2.6] (Characterization of the inclusion between bundle closures of
matrices). Let $A_{1},A_{2}\in{\mathbb{C}}^{n\times n}$. Then $\overline{\cal
B}(A_{2})\subseteq\overline{\cal B}(A_{1})$ if and only if
$A_{2}\in\overline{\cal O}(\psi_{c}(A_{1}))$, for some map
$\psi:{\mathbb{C}}\rightarrow{\mathbb{C}}$.

The statement of Theorem 20 is implicitly assuming that
$A_{2}\in\overline{\cal B}(A_{1})$ implies $\overline{{\cal
B}}(A_{2})\subseteq\overline{\cal B}(A_{1})$ (the counterpart of Lemma 11 for
matrices under similarity), but this can be proved using similar arguments to
the ones in the proof of Lemma 11, replacing matrix pencils by matrices, the
KCF by the JCF, and Theorem 7 by the corresponding one for orbits of matrices
under similarity, which only includes condition (iii) with $h=0$. Also, using
similar arguments to the ones in the proof of Theorem 16, including the fact
that the number of different bundles of $n\times n$ matrices is finite (the
counterpart of Lemma 13 for bundles of matrices under similarity), we can
prove the counterpart of Theorem 16 for bundles of matrices

######  Theorem 21.

Let $A\in{\mathbb{C}}^{n\times n}$. Then, there is a finite number of
different matrices, $A_{1},\ldots,A_{\ell}\in{\mathbb{C}}^{n\times n}$ (with,
say, $A_{1}=A$), such that

$\overline{\cal B}(A)=\bigcup_{i=1}^{\ell}{\cal B}(A_{i}).$

Moreover, $\overline{\cal B}(A_{i})$ is strictly included in $\overline{\cal
B}(A)$, for $i\neq 1$.

Finally, from Theorem 21 and using similar arguments to the ones in the proof
of Theorem 17, we can prove that bundles of matrices under similarity are open
in their closure:

######  Theorem 22.

Let $A\in{\mathbb{C}}^{n\times n}$. Then ${\cal B}(A)$ is an open set in its
closure.

###  4.2 Bundles of matrix polynomials of higher degree

From the previous developments, we can also conclude that bundles of matrix
polynomials are open in their closure.

A matrix polynomial of degree $d$ in the variable $\lambda$, say $P(\lambda)$,
is of the form $P(\lambda)=\sum_{i=0}^{d}\lambda^{i}A_{i},$ with
$A_{i}\in{\mathbb{C}}^{m\times n}$, for $0\leq i\leq d$, and $A_{d}\neq 0$. If
we allow $A_{d}$ to be zero, then we say that $P(\lambda)$ has grade $d$.
Matrix pencils are particular cases of matrix polynomials (namely, when
$d=1$).

We denote by ${\rm POL}_{d}^{m\times n}$ the set of complex matrix polynomials
of size $m\times n$ and grade $d$.

The orbit of a matrix polynomial $P(\lambda)$ is the subset of matrix
polynomials in ${\rm POL}_{d}^{m\times n}$ having exactly the same complete
spectral information as $P(\lambda)$ (see [11, p. 217]), i. e., the same
eigenvalues with the same partial multiplicities, and the same left and right
minimal indices. The definitions of these concepts can be found, for instance,
in [11] and the references therein. Similarly, the bundle of $P(\lambda)$,
denoted by ${\cal B}(P)$, is the subset of matrix polynomials in ${\rm
POL}_{d}^{m\times n}$ having the same spectral information as $P(\lambda)$,
“except that the values of the distinct eigenvalues are unspecified as long as
they remain distinct” (this sentence is taken from [8], for the case of
symmetric matrix polynomials). We want to emphasize that, for matrix
polynomials of higher grade, the action of strict equivalence is not
appropriate for defining orbits in this context, since two matrix polynomials
of grade larger than $1$ can have the same spectral information without being
strictly equivalent.

If ${\cal C}_{P}(\lambda)$ denotes the first Frobenius companion linearization
of $P(\lambda)$ (see, for instance, [11]), then $Q(\lambda)\in{\rm
POL}_{d}^{m\times n}$ belongs to the bundle of $P(\lambda)$ if and only if
${\cal C}_{Q}(\lambda)\in{\cal B}({\cal C}_{P})$. This is because $Q(\lambda)$
has the same complete spectral information as $P(\lambda)$ if and only if
${\cal C}_{P}(\lambda)$ has the same KCF as ${\cal C}_{Q}(\lambda)$. Moreover,
there is a homeomorphism,

$\begin{array}[]{cccc}f:&{\rm POL}_{d}^{m\times n}&\rightarrow&{\rm
GSYL}_{d}^{m\times n}\\\ &P(\lambda)&\mapsto&{\cal
C}_{P}(\lambda),\end{array}$ (23)

where ${\rm GSYL}_{d}^{m\times n}$ denotes the set of complex matrix pencils
of the form ${\cal C}_{P}(\lambda)$, with $P(\lambda)\in{\rm POL}_{d}^{m\times
n}$ (the so-called generalized Sylvester space in [11]). The fact that (23) is
a homeomorphism is immediate because it is an isometry (for instance, with the
following distance for matrix pencils and matrix polynomials, defined in terms
of the Frobenius norms of the coefficients,
$\rho(\sum_{i=0}^{d}\lambda^{i}A_{i},\sum_{i=0}^{d}\lambda^{i}A_{i}^{\prime}):=\left(\sum_{i=0}^{d}\|A_{i}-A_{i}^{\prime}\|_{F}\right)^{1/2}$,
see [11]).

Now, we define (see Eq. (5.4) in [8] for symmetric matrix polynomials and the
congruence relation)

${\cal B}^{\rm syl}({\cal C}_{P}):={\cal B}({\cal C}_{P})\cap{\rm
GSYL}_{d}^{m\times n}.$

Then, ${\cal B}^{\rm syl}({\cal C}_{P})$ is also open in its closure. This is
a consequence of ${\cal B}({\cal C}_{P})$ being open in its closure (Theorem
17). More precisely, one way to see this is the following: Let $M\in{\cal
B}^{\rm syl}({\cal C}_{P})={\cal B}({\cal C}_{P})\cap{\rm GSYL}_{d}^{m\times
n}$. Then, $M(\lambda)\in{\cal B}({\cal C}_{P})$, so there is a neighborhood
of $M(\lambda)$, say $U_{M}$, such that $U_{M}\cap\overline{\cal B}({\cal
C}_{P})\subseteq{\cal B}({\cal C}_{P})$. Now

$\begin{array}[]{ccl}U_{M}\cap\overline{\cal B}^{\rm syl}({\cal
C}_{P})&=&U_{M}\cap\overline{{\cal B}({\cal C}_{P})\cap{\rm GSYL}_{d}^{m\times
n}}\subseteq U_{M}\cap\overline{\cal B}({\cal C}_{P})\cap{\rm
GSYL}_{d}^{m\times n}\\\ &\subseteq&{\cal B}({\cal C}_{P})\cap{\rm
GSYL}_{d}^{m\times n}={\cal B}^{\rm syl}({\cal C}_{P}),\end{array}$

so there is also a neighborhood of $M(\lambda)$ whose intersection with
$\overline{\cal B}^{\rm syl}({\cal C}_{P})$ is contained in ${\cal B}^{\rm
syl}({\cal C}_{P})$, which means that ${\cal B}^{\rm syl}({\cal C}_{P})$ is
open in its closure.

Since (23) is a homeomorphism, we have

$f^{-1}({\cal B}^{\rm syl}({\cal C}_{P}))={\cal B}(P),\quad\mbox{and}\quad
f^{-1}\left(\overline{\cal B}^{\rm syl}({\cal C}_{P})\right)=\overline{\cal
B}(P).$

(see [8, p. 1047] for symmetric matrix polynomials). As a consequence, the
property of ${\cal B}^{\rm syl}({\cal C}_{P})$ being open in its closure can
be translated to ${\cal B}(P)$. More precisely, let $M(\lambda)\in{\cal
B}(P)$. Then $f(M)={\cal C}_{M}(\lambda)\in{\cal B}^{\rm syl}({\cal C}_{P})$,
and, as we have just seen, there is an open neighborhood of ${\cal C}_{M}$,
say $U_{M}$, such that $U_{M}\cap\overline{\cal B}^{\rm syl}({\cal
C}_{P})\subseteq{\cal B}^{\rm syl}({\cal C}_{P})$. Since $f$ is a
homeomorphism, $f^{-1}(U_{M})$ is an open neighborhood of $M$, say
$\widetilde{U}_{M}$, so that

$f^{-1}\left(U_{M}\cap\overline{\cal B}^{\rm syl}({\cal
C}_{P})\right)=\widetilde{U}_{M}\cap f^{-1}(\overline{\cal B}^{\rm syl}({\cal
C}_{P}))=\widetilde{U}_{M}\cap\overline{\cal B}(P),$

and

$f^{-1}(U_{M}\cap\overline{\cal B}^{\rm syl}({\cal C}_{P}))\subseteq
f^{-1}({\cal B}^{\rm syl}({\cal C}_{P}))={\cal B}(P)$

together imply that $\widetilde{U}_{M}\cap\overline{\cal B}(P)\subseteq{\cal
B}(P)$, for every $M(\lambda)\in{\cal B}(P)$, so ${\cal B}(P)$ is indeed open
in its closure.

We have then proved the following result.

######  Theorem 23.

Let $P(\lambda)\in{\rm POL}_{d}^{m\times n}$. Then, the bundle of $P(\lambda)$
(namely, the set of matrix polynomials in ${\rm POL}_{d}^{m\times n}$ with the
same complete spectral information as $P(\lambda)$, up to the specific values
of the distinct eigenvalues) is an open set in its closure.

##  5 Conclusions and open questions

This paper is mainly devoted to bundles of matrix pencils under strict
equivalence. We have provided necessary and sufficient conditions for the
closure of a given bundle to be included in the closure of another one
(Theorem 12). A proof that bundles are open in their closure is also given
(see Theorem 17). The same has been done for bundles of matrices under
similarity (in Theorem 22) and bundles of matrix polynomials of higher degree
(in Theorem 23). We have also revisited some notions already present in the
literature (like the one of “coalescence”), as well as some other previously
known results. Some additional technical results, that can be useful in the
future, have been proved.

Some further lines of research that naturally arise as a continuation of the
present work are the following:

  * •

We believe that, if $L(\lambda)$ and $L_{i}(\lambda)$ are as in Theorem 16,
then $\dim{\cal B}(L_{i})<\dim{\cal B}(L)$ for $i\neq 1$, with

$\dim{\cal B}(L):=\dim{\cal O}(L)+\\#\\{\mbox{different eigenvalues of
$L(\lambda)$}\\}$

(see, for instance, [16]). This would formally prove that $\overline{{\cal
B}}(L)$ is the union of ${\cal B}(L)$ together with a finite number of bundles
with smaller dimension. A natural approach to prove this fact is by using the
breakdown of the codimension count of pencil orbits provided in [5].

  * •

Given two matrix polynomials, $P_{1}(\lambda)$ and $P_{2}(\lambda)$, provide
necessary and sufficient conditions for $\overline{\cal
B}(P_{2})\subseteq\overline{\cal B}(P_{1})$.

  * •

Given two structured matrix pencils (or, more in general, matrix polynomials
of higher degree) $L_{1}(\lambda)$ and $L_{2}(\lambda)$, for any of the
structures mentioned in Section 1 (namely, alternating, (skew-) Hermitian,
(anti-) palindromic, or (skew-) symmetric), provide necessary and sufficient
conditions for $\overline{\cal B}(L_{2})\subseteq\overline{\cal B}(L_{1})$.
The bundles in this case should be defined for congruence or $*$-congruence,
but the definition in this case will probably deserve a more detailed care,
because of the restrictions and symmetries in the spectral information that
are imposed by the structure (see, for instance, [26]).

  * •

To prove whether or not bundles of structured matrix pencils (or matrix
polynomials of higher degree) are open in their closure. This is know to be
true, for instance, for the bundles of generic symmetric matrix pencils and
matrix polynomials with bounded rank, described in [8].

Acknowledgments. The authors are very much indebted to Inmaculada de Hoyos and
Juan Miguel Gracia for providing the source [3], that has been key to prove
Theorem 8 and for helpful discussions on this result, as well as for pointing
reference [25] out to us.

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