tcb@breakable

# Even grade generic skew-symmetric matrix polynomials with bounded rank

Fernando De Terán<EMAIL_ADDRESS>Andrii Dmytryshyn
<EMAIL_ADDRESS>Froilán M. Dopico<EMAIL_ADDRESS>Departamento
de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad
30, 28911, Leganés, Spain.  School of Science and Technology, Örebro
University, 701 82, Örebro, Sweden.

###### Abstract

We show that the set of $m\times m$ complex skew-symmetric matrix polynomials
of even grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$
is the closure of the single set of matrix polynomials with certain,
explicitly described, complete eigenstructure. This complete eigenstructure
corresponds to the most generic $m\times m$ complex skew-symmetric matrix
polynomials of even grade $d$ and rank at most $2r$. The analogous problem for
the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

###### keywords:

complete eigenstructure, genericity, matrix polynomials, skew-symmetry, normal
rank, orbits, pencils

###### MSC:

15A18, 15A21

##  1 Introduction

During the recent years, the problem of determining the generic
eigenstructures for sets of matrix pencils and matrix polynomials has been a
subject of research interest, in particular, for the sets of matrix pencils
and matrix polynomials with fixed grade and bounded rank. The description of
such sets as a union of closures of certain “generic sets” of pencils and
matrix polynomials is given in terms of their eigenstructures for general
matrix pencils [13] and polynomials [23], as well as for matrix pencils that
are skew-symmetric [24], symmetric [9], $T$-palindromic and $T$-alternating
[8], and Hermitian [11]. Moreover, in the case of odd grade the generic sets
are also derived for skew-symmetric [24] and symmetric [10] matrix
polynomials. In this paper, we tackle the even-grade-case for skew-symmetric
matrix polynomials. In Table 1 we present a summary of the contributions
mentioned above.

The reason for the even-grade-case being frequently omitted when structured
matrix polynomials are studied, e.g., [10, 21, 24], is the lack of structured
linearization templates [18, 40, 41]. In this paper, we resolve this issue by
adding an extra term to a given matrix polynomial of even grade (thus making
it of odd grade) and tracking how this addition affects the eigenstructure of
the polynomial. The main result is that the set of skew-symmetric matrix
polynomials of even grade and bounded rank is the closure of a single set with
a certain complete eigenstructure which is completely analogous to the one for
the odd-grade-case described in [24].

| Pencils ($d=1$) | Polynomials ($d>1$)
---|---|---
General |  | De Terán and Dopico,
---
2008, [13]
# = $r+1$
De Terán, Dopico
and Landsberg,
2017, [16]
| Dmytryshyn and Dopico,
---
2017, [23]
# = $rd+1$
Skew-Symmetric |  | Dmytryshyn and Dopico,
---
2018, [24]
# = $1$
| Dmytryshyn and Dopico,
---
2018, [24]
odd $d$
# = $1$
even $d$, this paper
Symmetric |  | De Terán, Dmytryshyn
---
and Dopico,
2020, [9]
# = $\left\lfloor r/2\right\rfloor+1$
| De Terán, Dmytryshyn
---
and Dopico,
2020, [10]
odd $d$
# = $\left\lfloor r\,d/2\right\rfloor+1$
| T-(anti)palindromic
---
(or T-even and odd)
|  De Terán, 2018, [8]
---
# = $1$
open
(skew-)Hermitian |  | De Terán, Dmytryshyn
---
and Dopico,
2022, [11]
# = $\left\lfloor(r+3)/2\right\rfloor\left(\left\lfloor
r/2\right\rfloor+1\right)$
open
Table 1: Solved and open problems about generic eigenstructures of sets of
general and structured matrix polynomials with bounded rank and degree.
Notation: $r$ is the rank, $d$ is the grade, and # is the number of generic
eigenstructures.

One reason to study generic sets is the possible applications in the
investigation of the effect of low rank perturbations on the spectral
information of pencils and matrix polynomials [4, 5, 6, 12, 14, 15, 19, 20,
35, 36, 42]. We also refer the reader to the introductions of the papers cited
in Table 1 for more references and background information. Determining the
generic eigenstructures of particular sets of structured matrix pencils and
matrix polynomials (in particular, those with bounded rank) is also useful for
obtaining the stratification of structured matrix pencils and polynomials.
Describing the stratification of matrix pencils and polynomials has been
considered in some references by different authors, for several sets of matrix
pencils and polynomials, like [30] for general matrix pencils, [31] for
controllability and observability pairs, [27] for system pencils associated
with state-space systems, [38] for full rank matrix polynomials, and [26] for
general matrix polynomials, as well as for skew-symmetric matrix pencils [28]
and odd-grade polynomials [22].

To facilitate the reading of this paper and its comparison with previous
results, we keep its structure and style as close as possible to the paper
[24] that covers the odd-grade-case. In this respect, we also warn the reader
that several repetitions of background concepts and auxiliary results
appearing in [24] are unavoidable to keep this paper self-contained. Thus,
Section 2 includes some basic results on general and skew-symmetric pencils
and also a new description of the set of skew-symmetric matrix pencils with
rank at most $2w$ and fixed number of canonical blocks associated with the
infinite eigenvalue. Section 3 presents a number of preliminary known results
on skew-symmetric matrix polynomials and some new auxiliary results on orbits
of skew-symmetric matrix polynomials. In Section 4 we prove our main result
for generic skew-symmetric matrix polynomials of fixed even grade and fixed
rank, i.e., Theorem 4.3. Moreover, Theorem 4.3 is combined with the
corresponding result for odd-grade skew-symmetric matrix polynomials obtained
in [24] for presenting in Theorem 4.5 the general result, which is independent
of the parity of the grade. Finally, Section 5 includes the computation of the
codimension of the orbit of the generic skew-symmetric matrix polynomials of
fixed even grade and bounded rank, which requires a different approach to the
one used to obtain the corresponding result for the odd-grade-case considered
in [24]. The conclusions and future research on generic eigenstructures of
matrix polynomials are discussed in Section 6.

##  2 Skew-symmetric matrix pencils

In this paper we consider matrix pencils and matrix polynomials whose matrix
coefficients have complex entries namely, a matrix polynomial is of the form
$P(\lambda)=\sum_{i=0}^{d}\lambda^{i}A_{i}$, with $A_{i}\in\mathbb{C}^{n\times
n}$, and a matrix pencil is a matrix polynomial with $d=1$. For matrix pencils
we use calligraphic letters. For brevity, the dependence on $\lambda$ in
$P(\lambda)$ will be often omitted, specially in the proofs, and we will use
just $P$ for denoting a matrix polynomial, and similarly for matrix pencils.

The rank of a matrix polynomial $P$, denoted by $\operatorname{rank}P$, is the
largest size of a non-identically zero minor (see [33, Ch. XII, §3] for matrix
pencils).

###  2.1 Preliminaries

In this section we recall the Kronecker canonical form (KCF) of general matrix
pencils and the canonical form of skew-symmetric matrix pencils under
congruence. Define $\overline{\mathbb{C}}:=\mathbb{C}\cup\infty$. For each
$k=1,2,\ldots$, and for each $\mu\in\mathbb{C}$ define the $k\times k$
matrices

$J_{k}(\mu):=\begin{bmatrix}\mu&1&&\\\ &\mu&\ddots&\\\ &&\ddots&1\\\
&&&\mu\end{bmatrix},\qquad I_{k}:=\begin{bmatrix}1&&&\\\ &1&&\\\ &&\ddots&\\\
&&&1\end{bmatrix},$

and for each $k=0,1,\ldots$, define the $k\times(k+1)$ matrices

$F_{k}:=\begin{bmatrix}0&1&&\\\ &\ddots&\ddots&\\\ &&0&1\\\
\end{bmatrix},\qquad G_{k}:=\begin{bmatrix}1&0&&\\\ &\ddots&\ddots&\\\
&&1&0\\\ \end{bmatrix}.$

All non-specified entries of $J_{k}(\mu),I_{k},F_{k},$ and $G_{k}$ are zeros.

Two $m\times n$ matrix pencils $\lambda A-B$ and $\lambda C-D$ are called
strictly equivalent if and only if there are non-singular matrices $Q$ and $R$
such that $Q^{-1}AR=C$ and $Q^{-1}BR=D$. We also define the orbit of $\lambda
A-B$ under the action of the group $GL_{m}(\mathbb{C})\times
GL_{n}(\mathbb{C})$ on the space of all matrix pencils by strict equivalence
as follows:

$\operatorname{O}^{e}(\lambda A-B)=\\{Q^{-1}(\lambda A-B)R\ :\ Q\in
GL_{m}(\mathbb{C}),R\in GL_{n}(\mathbb{C})\\}.$

Theorem 2.1 introduces the Kronecker Canonical Form (KCF) of matrix pencils.

######  Theorem 2.1.

[33, Ch. XII, Sect. 4] Each $m\times n$ matrix pencil $\lambda A-B$ is
strictly equivalent to a direct sum, uniquely determined up to permutation of
summands, of pencils of the form

$\displaystyle{\cal E}_{k}(\mu)$ $\displaystyle:=\lambda
I_{k}-J_{k}(\mu),\text{ with}\mu\in\mathbb{C},\quad{\cal
E}_{k}(\infty):=\lambda J_{k}(0)-I_{k},$ $\displaystyle{\cal L}_{k}$
$\displaystyle:=\lambda G_{k}-F_{k},\quad\text{ and }\quad{\cal
L}_{k}^{T}:=\lambda G_{k}^{T}-F^{T}_{k}.$

This direct sum is called the KCF of $\lambda A-B$.

The blocks ${\cal E}_{k}(\mu)$ and ${\cal E}_{k}(\infty)$ are associated to
the finite and infinite eigenvalues, respectively, and all together form the
regular part of $\lambda A-B$. The blocks ${\cal L}_{k}$ and ${\cal
L}_{k}^{T}$ are associated to the right and left minimal indices, respectively
(see page 3.1), and all together form the singular part of $\lambda A-B$. The
number of blocks ${\cal L}_{k}$ (respectively, ${\cal L}_{k}^{T}$) in the KCF
of $\lambda A-B$ is equal to the dimension of the right (respectively, left)
rational null-space of $\lambda A-B$.

A pencil $\lambda A-B$ is skew symmetric if and only if $(\lambda
A-B)^{T}=-(\lambda A-B)$. An $n\times n$ matrix pencil $\lambda A-B$ is called
congruent to $\lambda C-D$ if there is a non-singular matrix $S$ such that
$S^{T}AS=C$ and $S^{T}BS=D$. We also define the congruence orbit of $\lambda
A-B$ under the action of the group $GL_{n}(\mathbb{C})$ on the space of all
skew-symmetric matrix pencils by congruence as follows:

$\operatorname{O}^{c}(\lambda A-B)=\\{S^{T}(\lambda A-B)S\ :\ S\in
GL_{n}(\mathbb{C})\\}.$

In Theorem 2.2 we recall the canonical form under congruence of skew-symmetric
matrix pencils, so called skew-symmetric KCF.

######  Theorem 2.2.

[43] Each skew-symmetric $n\times n$ matrix pencil $\lambda A-B$ is congruent
to a direct sum, uniquely determined up to permutation of summands, of pencils
of the form

$\displaystyle{\cal H}_{h}(\mu)$
$\displaystyle:=\lambda\begin{bmatrix}0&I_{h}\\\
-I_{h}&0\end{bmatrix}-\begin{bmatrix}0&J_{h}(\mu)\\\
-J_{h}(\mu)^{T}&0\end{bmatrix},\quad\mu\in\mathbb{C},$ $\displaystyle{\cal
K}_{k}$ $\displaystyle:=\lambda\begin{bmatrix}0&J_{k}(0)\\\
-J_{k}(0)^{T}&0\end{bmatrix}-\begin{bmatrix}0&I_{k}\\\ -I_{k}&0\end{bmatrix},$
$\displaystyle{\cal M}_{m}$ $\displaystyle:=\lambda\begin{bmatrix}0&G_{m}\\\
-G_{m}^{T}&0\end{bmatrix}-\begin{bmatrix}0&F_{m}\\\
-F_{m}^{T}&0\end{bmatrix}.$

Notably the block ${\cal M}_{0}$ is the $1\times 1$ zero matrix pencil.
Similarly to the KCF, the blocks ${\cal H}_{h}(\mu)$ and ${\cal K}_{k}$
correspond to the finite and infinite eigenvalues, respectively, and all
together form the regular part of $\lambda A-B$. The blocks ${\cal M}_{m}$
correspond to the right (column) and left (row) minimal indices, which are
equal in the case of skew-symmetric matrix pencils, and form the singular part
of $\lambda A-B$. Theorem 2.2 also shows that skew-symmetric matrix pencils
always have even rank.

For brevity, we will refer to the number of $\cal K$-blocks or $\cal M$-blocks
of a skew-symmetric matrix pencil to mean the number of blocks of the form
${\cal K}_{k}$ or ${\cal M}_{k}$, respectively, in the skew-symmetric KCF of
the pencil.

###  2.2 Generic skew-symmetric matrix pencils with bounded rank and fixed
number of blocks corresponding to the infinite eigenvalue.

In this section we find the most generic skew-symmetric matrix pencils with
rank bounded by a fixed value and with exactly $r$ ${\cal K}$-blocks in the
skew-symmetric KCF. First, we recall some definitions and results.

By $\operatorname{PENCIL}_{m\times n}$ we denote the space of all $m\times n$
matrix pencils. A distance in $\operatorname{PENCIL}_{m\times n}$ can be
defined with the Frobenius norm of complex matrices [34] as $d(\lambda
A-B,\lambda C-D):=\sqrt{\|A-C\|_{F}^{2}+\|B-D\|_{F}^{2}}$, which makes
$\operatorname{PENCIL}_{m\times n}$ to be a metric space. This metric allows
us to consider closures of subsets of $\operatorname{PENCIL}_{m\times n}$, in
particular, closures of orbits by strict equivalence, denoted by
$\overline{\operatorname{O}^{e}}(\lambda A-B)$. Using these concepts, Theorem
2.3, see [7, 30], describes all the possible changes in the KCF of a general
unstructured matrix pencil under arbitrarily small perturbations. If in the
KCF of a matrix pencil the blocks ${\cal X}$ are changed to the blocks ${\cal
Y}$ (${\cal X}$ and ${\cal Y}$ of the same size) we write ${\cal
X}\rightsquigarrow{\cal Y}$.

######  Theorem 2.3.

[7] Let ${\cal P}_{1}$ and ${\cal P}_{2}$ be two matrix pencils in KCF. Then,
$\overline{\operatorname{O}^{e}}({\cal
P}_{1})\supset\operatorname{O}^{e}({\cal P}_{2})$ if and only if ${\cal
P}_{1}$ can be obtained from ${\cal P}_{2}$ by changing canonical blocks of
${\cal P}_{2}$ after applying a sequence of rules, that can be of the
following six types:

  1. 1.

${\cal L}_{j-1}\oplus{\cal L}_{k+1}\rightsquigarrow{\cal L}_{j}\oplus{\cal
L}_{k}$, $1\leqslant j\leqslant k;$

  2. 2.

${\cal L}_{j-1}^{T}\oplus{\cal L}_{k+1}^{T}\rightsquigarrow{\cal
L}_{j}^{T}\oplus{\cal L}_{k}^{T}$, $1\leqslant j\leqslant k;$

  3. 3.

${\cal L}_{j}\oplus{\cal E}_{k+1}(\mu)\rightsquigarrow{\cal
L}_{j+1}\oplus{\cal E}_{k}(\mu)$, $j,k=0,1,2,\dots$ and
$\mu\in\overline{\mathbb{C}};$

  4. 4.

${\cal L}_{j}^{T}\oplus{\cal E}_{k+1}(\mu)\rightsquigarrow{\cal
L}_{j+1}^{T}\oplus{\cal E}_{k}(\mu)$, $j,k=0,1,2,\dots$ and
$\mu\in\overline{\mathbb{C}};$

  5. 5.

${\cal E}_{j}(\mu)\oplus{\cal E}_{k}(\mu)\rightsquigarrow{\cal
E}_{j-1}(\mu)\oplus{\cal E}_{k+1}(\mu)$, $1\leqslant j\leqslant k$ and
$\mu\in\overline{\mathbb{C}};$

  6. 6.

${\cal L}_{p}\oplus{\cal L}_{q}^{T}\rightsquigarrow\bigoplus_{i=1}^{t}{\cal
E}_{k_{i}}(\mu_{i})$, if $p+q+1=\sum_{i=1}^{t}k_{i}$ and
$\mu_{i}\neq\mu_{i^{\prime}}$ for $i\neq
i^{\prime},\mu_{i}\in\overline{\mathbb{C}}.$

Observe that in the rules above any block ${\cal E}_{0}(\mu)$ should be
understood as the empty matrix.

The vector space of skew-symmetric matrix pencils of size $n\times n$ is
denoted by $\operatorname{PENCIL}_{n\times n}^{ss}$. A distance in
$\operatorname{PENCIL}_{n\times n}^{ss}$ is defined as in
$\operatorname{PENCIL}_{n\times n}$ and, thus, the topology considered in
$\operatorname{PENCIL}_{n\times n}^{ss}$ is the induced topology from
$\operatorname{PENCIL}_{n\times n}$. With this topology, we also consider
closures of subsets in $\operatorname{PENCIL}_{n\times n}^{ss}$ and, in
particular, closures of orbits by congruence of skew-symmetric matrix pencils,
which are denoted by $\overline{\operatorname{O}^{c}}(\lambda A-B)$.

We often use expressions as “the pencil ${\cal P}_{1}$ is more generic than
the pencil ${\cal P}_{2}$”, whose meaning is that
$\overline{\operatorname{O}^{e}}({\cal
P}_{1})\supset\operatorname{O}^{e}({\cal P}_{2})$ or
$\overline{\operatorname{O}^{c}}({\cal
P}_{1})\supset\operatorname{O}^{c}({\cal P}_{2})$, depending on the context.
In this language, the most generic skew-symmetric pencil with rank at most
$2w$ and with exactly $r$ ${\cal K}$-blocks associated to the infinite
eigenvalue in the skew-symmetric KCF, is a pencil such that the closure of its
congruence orbit includes the congruence orbit of any other skew-symmetric
pencil with rank at most $2w$ and with exactly $r$ ${\cal K}$-blocks. Define
$\operatorname{INF}_{r}\subset\operatorname{PENCIL}_{n\times n}^{ss}$ as the
set of all $n\times n$ skew-symmetric matrix pencils having exactly $r$ ${\cal
K}$-blocks.

######  Theorem 2.4.

Let $n,w$, and $r$ be integers such that $n\geq 2$, $2\leq 2w\leq n-1$ and
$r\leq w$. The set of $n\times n$ complex skew-symmetric matrix pencils with
rank at most $2w$ and with exactly $r$ ${\cal K}$-blocks corresponding to the
infinite eigenvalue in its skew-symmetric KCF, is a closed subset of
$\operatorname{INF}_{r}$ equal to $\overline{\operatorname{O}^{c}}({\cal
W})\cap\operatorname{INF}_{r}$, where

${\cal W}=\operatorname{diag}(\underbrace{{\cal M}_{\alpha+1},\dots,{\cal
M}_{\alpha+1}}_{s},\underbrace{{\cal M}_{\alpha},\dots,{\cal
M}_{\alpha}}_{n-2w-s},\underbrace{{\cal K}_{1},\dots,{\cal K}_{1}}_{r})\,$ (1)

with $\alpha=\lfloor(w-r)/(n-2w)\rfloor$ and
$s\equiv(w-r)\,\mathrm{mod}\,(n-2w)$.

###### Proof.

We follow a strategy similar to the one of the proof of [24, Thm. 3.1], but
taking into account the presence of $r$ ${\cal K}$-blocks, which were not
present in [24, Thm. 3.1]. More precisely, taking into account [28, Lem. 3.8]
(or the stronger result [28, Thm. 3.1]), in this proof we work with the KCF
rather than with the skew-symmetric KCF under congruence of Theorem 2.2, but
we always apply rules from Theorem 2.3 in pairs, such that the corresponding
change of the skew-symmetric KCF preserves transparently the skew-symmetry.

Note that the most generic skew-symmetric pencil with rank at most $2w$ and
exactly $r$ ${\cal K}$ blocks has rank exactly $2w$, because otherwise there
would be more than $n-2w$ pairs of equal left and right minimal indices
(equivalently, in the skew-symmetric KCF there would be more than $n-2w$
${\cal M}$-blocks, since the number of left/right singular blocks is equal to
the dimension of the rational left/right null-space of the pencil). Moreover,
due to the even parity of the rank, there would be an excess of at least two
pairs of equal left and right minimal indices. In that case, the rule 6 in
Theorem 2.3 can be applied twice to obtain a more generic skew-symmetric
matrix pencil with rank larger in two units and with exactly $r$ ${\cal
K}$-blocks. Therefore, we focus, in the rest of the proof, on $n\times n$
skew-symmetric matrix pencils with rank exactly $2w$ and with exactly $r$
${\cal K}$-blocks. Each of such pencils has the following KCF:

$\begin{split}\operatorname{diag}({\cal L}_{\gamma_{1}},\dots,{\cal
L}_{\gamma_{n-2w}},&{\cal L}^{T}_{\gamma_{1}},\dots,{\cal
L}^{T}_{\gamma_{n-2w}},{\cal J},{\cal J},\\\ &{\cal
E}_{k_{1}}(\infty),\dots,{\cal E}_{k_{r}}(\infty),{\cal
E}_{k_{1}}(\infty),\dots,{\cal E}_{k_{r}}(\infty)),\end{split}$

since the left and right singular blocks are paired up according to Theorem
2.2 and the blocks ${\cal J}$ of the regular part associated with the finite
eigenvalues are paired up, as well as the blocks that correspond to the
infinite eigenvalue.

Note that the sizes of the $2r$ blocks that correspond to the infinite
eigenvalue, i.e. ${\cal E}_{k_{i}}(\infty)$, in the KCF of a generic skew-
symmetric matrix pencil of rank equal to $2w$ and with exactly $r$ ${\cal
K}$-blocks can not be larger than 1 (otherwise rules 3 and 4 from Theorem 2.3
can be applied to obtain a more generic skew-symmetric pencil and to decrease
the sizes of the $2r$ blocks ${\cal E}_{k_{i}}(\infty)$ to $1$). The remaining
part of the proof repeats the proof of [24, Thm. 3.1] with the difference that
[23, Thm. 2.7] is now used for determining the most generic
$(w-r)\times(n-r-w)$ matrix pencil and [23, Thm. 2.8] for determining the most
generic $(n-r-w)\times(w-r)$ matrix pencil. The details can be found in [24],
but the proof reduces to apply rules 3 and 4 from Theorem 2.3 for every couple
of identical single Jordan blocks associated with a finite eigenvalue (using a
left and a right singular block ${\cal L}_{j}$ and ${\cal L}_{j}^{T}$, which
are coupled) to eliminate these Jordan blocks by shrinking the size to $0$,
and then applying rules 1 and 2 from Theorem 2.3 to each couple of blocks
${\cal L}_{j-1}\oplus{\cal L}_{k+1}$ and ${\cal L}_{j-1}^{T}\oplus{\cal
L}_{k+1}^{T}$. ∎

##  3  Auxiliary results on skew-symmetric matrix polynomials

###  3.1 Complete eigenstructure of skew-symmetric matrix polynomials

We consider skew-symmetric $m\times m$ matrix polynomials $P(\lambda)$ of
grade $d$, i.e., of degree less than or equal to $d$, over $\mathbb{C}$:

$P(\lambda)=\lambda^{d}A_{d}+\dots+\lambda A_{1}+A_{0},\quad
A_{i}^{T}=-A_{i},\ A_{i}\in\mathbb{C}^{m\times m}\ \text{for }i=0,\dots,d.$

Note that we do not require that the leading coefficient, $A_{d}$, of
$P(\lambda)$ is nonzero. As usual, the degree of $P(\lambda)$, denoted as
$\deg(P)$, is the largest index $k$ such that $A_{k}\neq 0$. We denote the
vector space of $m\times m$ skew-symmetric matrix polynomials of grade $d$ by
$\operatorname{POL}_{d,m\times m}^{ss}$. We write $\operatorname{POL}$ instead
of $\operatorname{POL}_{d,m\times m}^{ss}$, if there is no risk of confusion.
Note that $\operatorname{POL}_{1,n\times
n}^{ss}=\operatorname{PENCIL}_{n\times n}^{ss}$. As in the case of pencils,
see Sections 2, by using the Frobenius matrix norm of complex matrices [34], a
distance in $\operatorname{POL}_{d,m\times m}^{ss}$ is defined as
$d(P,P^{\prime})=\left(\sum_{i=0}^{d}||A_{i}-A^{\prime}_{i}||_{F}^{2}\right)^{\frac{1}{2}}$,
where $P(\lambda)=\sum_{i=0}^{d}\lambda^{i}A_{i}$ and
$P^{\prime}(\lambda)=\sum_{i=0}^{d}\lambda^{i}A^{\prime}_{i}$, making
$\operatorname{POL}_{d,m\times m}^{ss}$ a metric space with the Euclidean
topology induced by this distance. For convenience, we define the Frobenius
norm of the matrix polynomial $P$ as
$||P(\lambda)||_{F}=\left(\sum_{i=0}^{d}||A_{i}||_{F}^{2}\right)^{\frac{1}{2}}$.
An arbitrarily small pencil or matrix polynomial is a pencil or matrix
polynomial with arbitrarily small Frobenius norm.

Two matrix polynomials $P(\lambda)$ and $Q(\lambda)$ are called unimodularly
congruent if $F(\lambda)^{T}P(\lambda)F(\lambda)=Q(\lambda)$ for some
unimodular matrix polynomial $F(\lambda)$ (i.e. $\det
F(\lambda)\in\mathbb{C}\backslash\\{0\\}$), see also [41].

######  Theorem 3.1.

[41] Let $P(\lambda)$ be a skew-symmetric $m\times m$ matrix polynomial. Then
there exist $r\in\mathbb{N}$ with $2r\leqslant m$ and a unimodular matrix
polynomial $F(\lambda)$ such that

$F(\lambda)^{T}P(\lambda)F(\lambda)=\begin{bmatrix}0&g_{1}(\lambda)\\\
-g_{1}(\lambda)&0\end{bmatrix}\oplus\dots\oplus\begin{bmatrix}0&g_{r}(\lambda)\\\
-g_{r}(\lambda)&0\end{bmatrix}\oplus 0_{m-2r}=:S(\lambda),$

where $g_{j}$ is a monic polynomial, for $j=1,\dots,r$, and $g_{j}(\lambda)$
divides $g_{j+1}(\lambda)$, for $j=1,\dots,r-1$. Moreover, the canonical form
$S(\lambda)$ is unique.

Similarly to the unimodular congruence, two matrix polynomials $P(\lambda)$
and $Q(\lambda)$ are called unimodularly equivalent if
$U(\lambda)P(\lambda)V(\lambda)=Q(\lambda)$ for some unimodular matrix
polynomials $U(\lambda)$ and $V(\lambda)$ (i.e. $\det U(\lambda),\det
V(\lambda)\in\mathbb{C}\backslash\\{0\\}$). Notably, the canonical form in
Theorem 3.1 is the skew-symmetric version of the well-known Smith form for
matrix polynomials under unimodular equivalence [33, 41].

Note that the (normal) rank of the skew-symmetric matrix polynomial
$P(\lambda)$ in Theorem 3.1 is equal to the nonnegative integer $2r$, i.e., it
is always an even number. The monic scalar polynomials
$g_{1}(\lambda),\dots,g_{r}(\lambda)$ in Theorem 3.1 are called the invariant
polynomials of $P(\lambda)$, and, for any $\alpha\in\mathbb{C}$, each of them
can be uniquely factored as

$g_{j}(\lambda)=(\lambda-\alpha)^{\sigma_{j}}p_{j}(\lambda),\quad\mbox{with
$p_{j}(\alpha)\neq 0$}$

and $\sigma_{j}\geqslant 0$ being an integer. The sequence
$0\leq\sigma_{1}=\sigma_{1}\leq\sigma_{2}=\sigma_{2}\leq\cdots\leq\sigma_{r}=\sigma_{r}$
is called the sequence of partial multiplicities of $P(\lambda)$ at $\alpha$.
The number $\alpha$ is a finite eigenvalue of $P(\lambda)$ if the partial
multiplicity sequence at $\alpha$ contains at least two nonzero terms, or,
equivalently, if $\alpha$ is a root of at least one invariant polynomial
$g_{j}(\lambda)$. The elementary divisors of $P(\lambda)$ associated with a
finite eigenvalue $\alpha$ is the collection of factors
$(\lambda-\alpha)^{\sigma_{1}},(\lambda-\alpha)^{\sigma_{1}},(\lambda-\alpha)^{\sigma_{2}},(\lambda-\alpha)^{\sigma_{2}},\ldots,(\lambda-\alpha)^{\sigma_{r}},(\lambda-\alpha)^{\sigma_{r}}$
for which $\sigma_{j}>0$.

Let $P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$. Then, the sequence
of partial multiplicities of $P(\lambda)$ at infinity is defined to be the
sequence of partial multiplicities of the matrix polynomial
$\operatorname{rev}_{d}P(\lambda):=\lambda^{d}P(1/\lambda)$ at zero. Moreover,
if zero is an eigenvalue of $\operatorname{rev}_{d}P(\lambda)$, then we say
that $\lambda=\infty$ is an eigenvalue of the matrix polynomial
$P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$. The elementary divisors
for the zero eigenvalue of $\operatorname{rev}_{d}P(\lambda)$ are the
elementary divisors associated with the infinite eigenvalue of $P(\lambda)$.
We emphasize that the sequence of partial multiplicities of $P(\lambda)$ at
infinity, as well as the fact that $\lambda=\infty$ is an eigenvalue of
$P(\lambda)$, depend on the grade $d$ chosen for $P(\lambda)$. This
observation plays a key role in the results of this paper.

Theorem 3.2 connects the smallest partial multiplicity at $\infty$ with the
grade and the degree of the matrix polynomial and with the fact of the leading
coefficient being equal to zero. It is a consequence of [2, Lemma 2.6]. The
complete proof can be found in [1, Lemma 2.7]. Since Theorem 3.2 is important
in this paper we sketch the simple proof for the sake of completeness.

######  Theorem 3.2.

Let $P(\lambda)=\sum_{i=0}^{d}\lambda^{i}P_{i}\in\operatorname{POL}_{d,m\times
m}^{ss}$ with $\operatorname{rank}P=2r,\ r>0,$ and sequence of partial
multiplicities at $\infty$ equal to
$0\leq\gamma_{1}=\gamma_{1}\leq\gamma_{2}=\gamma_{2}\leq\dots\leq\gamma_{r}=\gamma_{r}$.
Then,

  * (a)

$\gamma_{1}=d-\deg(P)$, and

  * (b)

$P_{d}=0$ if and only if
$1\leq\gamma_{1}=\gamma_{1}\leq\gamma_{2}=\gamma_{2}\leq\dots\leq\gamma_{r}=\gamma_{r}$.

###### Proof.

Observe that $P_{d}=0$ if and only if $d>\deg(P)$. Therefore part (b) follows
from part (a), so we only need to prove part (a). Note that
$\operatorname{rev}_{d}P(\lambda)=\lambda^{d-\deg(P)}\operatorname{rev}_{\deg(P)}P(\lambda)$.
Therefore, the (skew-symmetric) Smith form of
$\operatorname{rev}_{d}P(\lambda)$ is the one of
$\operatorname{rev}_{\deg(P)}P(\lambda)$ multiplied by $\lambda^{d-\deg(P)}$,
which implies that the smallest partial multiplicity of $P$ at $\infty$ is
$\gamma_{1}=d-\deg(P)+\widetilde{\gamma}_{1}$, where $\widetilde{\gamma}_{1}$
is the smallest partial multiplicity at $0$ of
$\operatorname{rev}_{\deg(P)}P(\lambda)$. The fact that
$(\operatorname{rev}_{\deg(P)}P)(0)=P_{\deg(P)}\neq 0$, implies that
$\widetilde{\gamma}_{1}=0$ and the result is proved. ∎

For an $m\times n$ matrix polynomial $P(\lambda)$, the left and right null-
spaces, over the field of rational functions $\mathbb{C}(\lambda)$, are
defined as follows:

$\displaystyle{\cal N}_{\rm left}(P)$
$\displaystyle:=\\{y(\lambda)^{T}\in\mathbb{C}(\lambda)^{1\times
m}:y(\lambda)^{T}P(\lambda)=0_{1\times n}\\},$ $\displaystyle{\cal N}_{\rm
right}(P)$ $\displaystyle:=\\{x(\lambda)\in\mathbb{C}(\lambda)^{n\times
1}:P(\lambda)x(\lambda)=0_{m\times 1}\\}.$

Each subspace ${\cal V}$ of $\mathbb{C}(\lambda)^{n}$ has bases consisting
entirely of vector polynomials. A basis of ${\cal V}$ consisting of vector
polynomials whose sum of degrees is minimal among all bases of ${\cal V}$
consisting of vector polynomials is called a minimal basis of ${\cal V}$. The
ordered list of degrees of the vector polynomials in any minimal basis of
${\cal V}$ is always the same. These degrees are called the minimal indices of
${\cal V}$ [32, 39]. This allows us to define the left and right minimal
indices of a matrix polynomial $P(\lambda)$ as those of ${\cal N}_{\rm
left}(P)$ and ${\cal N}_{\rm right}(P)$, respectively. Note that for a skew-
symmetric matrix polynomial the left minimal indices are equal to the right
ones.

We define the complete eigenstructure of a skew-symmetric matrix polynomial
$P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$ of grade $d$ to be the
collection of all finite and infinite eigenvalues, the corresponding
elementary divisors (or, equivalently, the corresponding sequences of partial
multiplicities), and the left and right minimal indices of $P(\lambda)$. To
refer concisely to the subset of $\operatorname{POL}^{ss}_{d,m\times m}$ of
skew-symmetric matrix polynomials with the same size $m\times m$, with the
same grade $d$, and with the same complete eigenstructure, we define the
notion of orbit of a skew-symmetric matrix polynomial. Notably, an analogous
definition for general polynomials is given in [23].

######  Definition 3.3.

Let $P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$. The subset of matrix
polynomials in $\operatorname{POL}^{ss}_{d,m\times m}$ with the same complete
eigenstructure as $P$ is called the _orbit_ of $P$, denoted
$\operatorname{O}(P)$.

Observe that Theorem 3.2-(a) implies that all the polynomials in
$\operatorname{O}(P)$ have the same degree. Moreover, they also have the same
rank, since the rank is determined by the size $m\times m$ and the number of
left (or right) minimal indices. Note that, by contrast to congruence orbits
of skew-symmetric matrix pencils, Definition 3.3 is not associated with an
action of any group.

###  3.2 Orbits of the same skew-symmetric matrix polynomial viewed with
different grades

The proofs of the main results in this paper rely on comparing the properties
of a skew-symmetric matrix polynomial
$P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$ of grade $d$ with the
entry-wise identical polynomial
$\widetilde{P}(\lambda)=\lambda^{d+1}\,0+P(\lambda)\in\operatorname{POL}^{ss}_{d+1,m\times
m}$ of grade $d+1$, as well as on comparing the orbits, and their closures, of
these two matrix polynomials. Lemma 3.4 establishes these comparisons. Recall
in the statement of this lemma that
$\operatorname{O}(P)\subseteq\overline{\operatorname{O}}(P)\subseteq\operatorname{POL}^{ss}_{d,m\times
m}$ and that
$\operatorname{O}(\widetilde{P})\subseteq\overline{\operatorname{O}}(\widetilde{P})\subseteq\operatorname{POL}^{ss}_{d+1,m\times
m}$.

######  Lemma 3.4.

Let $P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$ have rank $2r,\ r>0,$
and let
$\widetilde{P}(\lambda):=\lambda^{d+1}\,0+P(\lambda)\in\operatorname{POL}^{ss}_{d+1,m\times
m}$. Then:

  * (a)

$\operatorname{rank}P=\operatorname{rank}\widetilde{P}=2r$.

  * (b)

$P(\lambda)$ and $\widetilde{P}(\lambda)$ have the same finite eigenvalues,
with the same sequences of partial multiplicities, or, equivalently, with the
same elementary divisors.

  * (c)

$P(\lambda)$ and $\widetilde{P}(\lambda)$ have the same left and right minimal
indices.

  * (d)

$\gamma_{1}=\gamma_{1}\leq\gamma_{2}=\gamma_{2}\leq\dots\leq\gamma_{r}=\gamma_{r}$
is the sequence of partial multiplicities at $\infty$ of $P(\lambda)$ if and
only if
$\gamma_{1}+1=\gamma_{1}+1\leq\gamma_{2}+1=\gamma_{2}+1\leq\dots\leq\gamma_{r}+1=\gamma_{r}+1$
is the sequence of partial multiplicities at $\infty$ of
$\widetilde{P}(\lambda)$.

  * (e)

If
$\widetilde{Q}(\lambda)=\sum_{i=0}^{d+1}\lambda^{i}Q_{i}\in\operatorname{O}(\widetilde{P})$,
then $Q_{d+1}=0$.

  * (f)

If
$\widetilde{Q}(\lambda)=\sum_{i=0}^{d+1}\lambda^{i}Q_{i}\in\overline{\operatorname{O}}(\widetilde{P})$,
then $Q_{d+1}=0$.

  * (g)

Let
$\widetilde{Q}(\lambda)=\lambda^{d+1}\,0+Q(\lambda)\in\operatorname{POL}^{ss}_{d+1,m\times
m}$, where $Q(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$. Then,

    1. (a)

$\widetilde{Q}\in\operatorname{O}(\widetilde{P})$ if and only if
$Q\in\operatorname{O}(P)$,

    2. (b)

$\widetilde{Q}\in\overline{\operatorname{O}}(\widetilde{P})$ if and only if
$Q\in\overline{\operatorname{O}}(P)$, and

    3. (c)

$\operatorname{O}(\widetilde{Q})\subseteq\overline{\operatorname{O}}(\widetilde{P})$
if and only if $\operatorname{O}(Q)\subseteq\overline{\operatorname{O}}(P)$.

###### Proof.

Parts (a), (b) and (c) are trivial because the rank, the (skew-symmetric)
Smith form and the left and right rational null spaces of a matrix polynomial
do not depend on the grade chosen for that polynomial.

Part (d) follows from the fact that
$\operatorname{rev}_{d+1}\widetilde{P}(\lambda)=\lambda\operatorname{rev}_{d}P(\lambda)$.
Therefore, the (skew-symmetric) Smith form of
$\operatorname{rev}_{d+1}\widetilde{P}(\lambda)$ is equal to $\lambda$ times
the Smith form of $\operatorname{rev}_{d}P(\lambda)$, which implies the
result.

Part (e). Since all the matrix polynomials in
$\operatorname{O}(\widetilde{P})\subseteq\operatorname{POL}^{ss}_{d+1,m\times
m}$ have the same complete eigenstructure, all of them have the same sequence
of partial multiplicities at $\infty$. Then, Theorem 3.2-(a) implies that all
of them have the same degree. So, $Q_{d+1}=0$ because the coefficient of
$\lambda^{d+1}$ in $\widetilde{P}$ is $P_{d+1}=0$.

Part (f).
$\widetilde{Q}(\lambda)=\sum_{i=0}^{d+1}\lambda^{i}Q_{i}\in\overline{\operatorname{O}}(\widetilde{P})$
if and only if $\widetilde{Q}$ is the limit of a sequence of matrix
polynomials in $\operatorname{O}(\widetilde{P})$. But, by (e), all the terms
in this sequence have the coefficient of $\lambda^{d+1}$ equal to zero. So,
the limit of the sequence have also the coefficient of $\lambda^{d+1}$ equal
to zero.

Part (g)-1 follows from the definition of orbit and from applying (b), (c),
and (d) to $P$ and $\widetilde{P}$ and to $Q$ and $\widetilde{Q}$.

Part (g)-2. If $\widetilde{Q}\in\overline{\operatorname{O}}(\widetilde{P})$,
then $\widetilde{Q}=\lim_{k\rightarrow\infty}\widetilde{Q}^{(k)}$, for a
sequence $\\{\widetilde{Q}^{(k)}\\}_{k\in\mathbb{N}}$ whose terms belong to
$\operatorname{O}(\widetilde{P})\subset\operatorname{POL}^{ss}_{d+1,m\times
m}$. Taking into account (e), we have that
$\widetilde{Q}^{(k)}(\lambda)=\lambda^{d+1}\,0+Q^{(k)}(\lambda)$, where
$\\{Q^{(k)}\\}_{k\in\mathbb{N}}\subset\operatorname{POL}^{ss}_{d,m\times m}$,
and, taking into account (g)-1,
$\\{Q^{(k)}\\}_{k\in\mathbb{N}}\subset\operatorname{O}(P)$. Thus,
$Q=\lim_{k\rightarrow\infty}Q^{(k)}$, which implies that
$Q\in\overline{\operatorname{O}}(P)$.

Conversely, if $Q\in\overline{\operatorname{O}}(P)$, then
$Q=\lim_{k\rightarrow\infty}Q^{(k)}$, for a sequence
$\\{Q^{(k)}\\}_{k\in\mathbb{N}}$ whose terms belong to
$\operatorname{O}(P)\subset\operatorname{POL}^{ss}_{d,m\times m}$. By part
(g)-1, the sequence $\\{\lambda^{d+1}\,0+Q^{(k)}\\}_{k\in\mathbb{N}}$ is
included in
$\operatorname{O}(\widetilde{P})\subset\operatorname{POL}^{ss}_{d+1,m\times
m}$. Moreover,
$\lim_{k\rightarrow\infty}\lambda^{d+1}\,0+Q^{(k)}=\widetilde{Q}$, which
implies that $\widetilde{Q}\in\overline{\operatorname{O}}(\widetilde{P})$.

Finally, for part (g)-3, let us first assume that
$\operatorname{O}(\widetilde{Q})\subseteq\overline{\operatorname{O}}(\widetilde{P}).$
Let $Q^{\prime}\in\operatorname{O}(Q)$. We want to prove that
$Q^{\prime}\in\overline{\operatorname{O}}(P)$. Since
$Q^{\prime}\in\operatorname{O}(Q)$, by part (g)-1 we conclude that
$\widetilde{Q}^{\prime}:=\lambda^{d+1}0+Q^{\prime}\in\operatorname{O}(\widetilde{Q})$,
and then
$\widetilde{Q}^{\prime}\in\overline{\operatorname{O}}(\widetilde{P})$, by
hypothesis. But now part (g)-2 implies
$Q^{\prime}\in\overline{\operatorname{O}}(P)$.

Conversely, let us assume that
$\operatorname{O}(Q)\subseteq\overline{\operatorname{O}}(P)$, and let
$\widetilde{Q}^{\prime}\in\operatorname{O}(\widetilde{Q})$. We want to prove
that $\widetilde{Q}^{\prime}\in\overline{\operatorname{O}}(\widetilde{P})$. By
part (e) we know that $\widetilde{Q}^{\prime}=\lambda^{d+1}0+Q^{\prime}$, for
some $Q^{\prime}\in{\rm POL}^{ss}_{d,m\times m}$, so part (g)-1 together with
the hypothesis $\widetilde{Q}^{\prime}\in\operatorname{O}(\widetilde{Q})$
imply that $Q^{\prime}\in\operatorname{O}(Q)$. But this in turn implies
$Q^{\prime}\in\overline{\operatorname{O}}(P)$, by hypothesis, so part (g)-2
gives $\widetilde{Q}^{\prime}\in\overline{\operatorname{O}}(\widetilde{P})$,
as wanted. ∎

###  3.3 Linearization of skew-symmetric matrix polynomials and their
perturbations

A matrix pencil ${\cal F}_{P}$ is called a linearization of a matrix
polynomial $P(\lambda)$ of grade $d$ if ${\cal F}_{P}$ has the same finite
eigenvalues and associated elementary divisors, the same number of left
minimal indices, and the same number of right minimal indices as $P$ [18, Thm.
4.1]. If in addition, $\operatorname{rev}_{1}{\cal F}_{P}$ is a linearization
of $\operatorname{rev}_{d}P$ then ${\cal F}_{P}$ is called a strong
linearization of $P$ and, then, ${\cal F}_{P}$ and $P$ have also the same
infinite elementary divisors.

The following pencil-template is known to be a skew-symmetric strong
linearization of skew-symmetric matrix polynomials
$P(\lambda)=\sum_{i=0}^{d}\lambda^{i}A_{i}\in\operatorname{POL}^{ss}_{d,m\times
m}$ of odd grade $d$ [41], see also [3, 40]:

$\displaystyle{{\cal F}}_{P}(i,i)$ $\displaystyle=\begin{cases}\lambda
A_{d-i+1}+A_{d-i}&\text{if }i\text{ is odd,}\\\ 0&\text{if }i\text{ is
even,}\\\ \end{cases}$ $\displaystyle{{\cal F}}_{P}(i,i+1)$
$\displaystyle=\begin{cases}-I_{m}&\text{if }i\text{ is odd,}\\\ -\lambda
I_{m}&\text{if }i\text{ is even,}\\\ \end{cases}\quad{{\cal
F}}_{P}(i+1,i)=\begin{cases}I_{m}&\text{if }i\text{ is odd,}\\\ \lambda
I_{m}&\text{if }i\text{ is even,}\\\ \end{cases}$

where ${\cal F}_{P}(j,k)$ denotes an $m\times m$ matrix pencil which is at the
position $(j,k)$ of the block pencil ${\cal F}_{P}$ and $j,k=1,\dots,d$. The
blocks of ${\cal F}_{P}$ in positions which are not specified above are zero.
We rewrite this strong linearization template in a matrix form as follows:

${\cal F}_{P}(\lambda)=\lambda\begin{bmatrix}A_{d}&&&&&\\\
&\ddots&\ddots&&&\\\ &\ddots&0&-I&&\\\ &&I&A_{3}&&\\\ &&&&0&-I\\\
&&&&I&A_{1}\\\ \end{bmatrix}-\begin{bmatrix}-A_{d-1}&I&&&&\\\
-I&0&\ddots&&&\\\ &\ddots&\ddots&&&\\\ &&&-A_{2}&I&\\\ &&&-I&0&\\\
&&&&&-A_{0}\\\ \end{bmatrix}.$ (2)

Note that the linearization ${{\cal F}}_{P}$ is defined only for skew-
symmetric matrix polynomials of odd grade. The reason is that there is no
skew-symmetric linearization-template (i.e., a skew-symmetric companion form
in the language of [18, Sects. 5 and 7]) for skew-symmetric matrix polynomials
of even grade [18, 41].

For any skew-symmetric matrix polynomial
$P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$, the strong linearization
(2) preserves the finite and infinite elementary divisors of $P(\lambda)$ but
does not preserve the left and right minimal indices of $P(\lambda)$.
Nevertheless, the relations between the minimal indices of a skew-symmetric
matrix polynomial $P(\lambda)$ and its linearization (2) are derived in [22],
see also [17]. We recall them in Theorem 3.5.

######  Theorem 3.5.

[22] Let $P(\lambda)$ be a skew-symmetric $m\times m$ matrix polynomial of odd
grade $d\geqslant 3$, and let ${\cal F}_{P}$ be its strong linearization (2)
given above. If
$0\leqslant\varepsilon_{1}\leqslant\varepsilon_{2}\leqslant~{}\dots\leqslant\varepsilon_{t}$
are the right (left) minimal indices of $P$ then

$0\leqslant\varepsilon_{1}+\frac{1}{2}(d-1)\leqslant\varepsilon_{2}+\frac{1}{2}(d-1)\leqslant\cdots\leqslant\varepsilon_{t}+\frac{1}{2}(d-1)$

are the right (left) minimal indices of ${{\cal F}}_{P}$.

The linearization ${\cal F}_{P}$ in (2) is crucial for obtaining the results
in Section 4. Thus we define the generalized Sylvester space consisting of the
linearizations ${\cal F}_{P}$ of all $m\times m$ skew-symmetric matrix
polynomials of odd grade $d$, namely:

${\operatorname{GSYL}^{ss}_{d,m\times m}=\\{{\cal F}_{P}\
:P(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}\;\text{with odd}\;d\\}.}$

If there is no risk of confusion we will write $\operatorname{GSYL}$ instead
of $\operatorname{GSYL}^{ss}_{d,m\times m}$. Given ${\cal F}_{P}=\lambda A-B$
and ${\cal F}_{P^{\prime}}=\lambda A^{\prime}-B^{\prime}$, the function
$d({\cal F}_{P},{\cal
F}_{P^{\prime}}):=\left(||A-A^{\prime}||_{F}^{2}+||B-B^{\prime}||_{F}^{2}\right)^{\frac{1}{2}}$
mentioned in Sections 2.2 and 3.1 is a distance on $\operatorname{GSYL}$ and
it makes $\operatorname{GSYL}$ a metric space. Since $d({{\cal F}}_{P},{{\cal
F}}_{P^{\prime}})=d(P,P^{\prime})$, there is a bijective isometry (and
therefore a homeomorphism):

$f:\operatorname{POL}^{ss}_{d,m\times
m}\rightarrow\operatorname{GSYL}^{ss}_{d,m\times m}\quad\text{such
that}\quad{f(P)={\cal F}_{P}}.$

We also define the orbit of the skew-symmetric linearizations of type (2) of a
fixed skew-symmetric matrix polynomial $P\in\operatorname{POL}^{ss}_{d,m\times
m}$ of odd grade $d$ as

$\operatorname{O}({\cal F}_{P})=\\{(S^{T}{\cal
F}_{P}S)\in\operatorname{GSYL}^{ss}_{d,m\times m}\ :\ S\in
GL_{md}(\mathbb{C})\\}.$ (3)

We emphasize that all the elements of $\operatorname{O}({{\cal F}}_{P})$ have
the block structure of the elements of $\operatorname{GSYL}^{ss}_{d,m\times
m}$. Thus, in particular, $\operatorname{O}(P)=f^{-1}(\operatorname{O}({{\cal
F}}_{P}))$, as a consequence of the properties of strong linearizations and
Theorem 3.5, and
$\overline{\operatorname{O}}(P)=f^{-1}(\overline{\operatorname{O}}({{\cal
F}}_{P}))$, as a consequence of $f$ being a homeomorphism. Moreover, we also
have that, for any $m\times m$ skew-symmetric matrix polynomials $P,Q$ of odd
grade $d$,
$\overline{\operatorname{O}}(P)\supseteq\overline{\operatorname{O}}(Q)$ if and
only if $\overline{\operatorname{O}}({{\cal
F}}_{P})\supseteq\overline{\operatorname{O}}({{\cal F}}_{Q})$, where it is
essential to note that the closures are taken in the metric spaces
$\operatorname{POL}$ and $\operatorname{GSYL}$, respectively, defined above.
Similarly to the unstructured matrix pencil case, $\operatorname{O}({{\cal
F}}_{P})$ is open in its closure in the relative Euclidean topology, and so is
$\operatorname{O}(P)$ since $f$ is a homeomorphism.

##  4 Generic skew-symmetric matrix polynomials with bounded rank and fixed
grade

In this section we present the complete eigenstructure of the generic $m\times
m$ skew-symmetric matrix polynomials of rank at most $2r$ and even grade $d$
in Theorem 4.3. Since the case of odd grade is known, we also provide a result
that does not depend on the parity of $d$ in Theorem 4.5. We start by proving
the following auxiliary lemma.

######  Lemma 4.1.

Let $d,m$, and $r$ be integers such that $d\geq 1$, $m\geq 2$, and $2\leq
2r\leq(m-1)$. For any skew-symmetric matrix polynomial
$Q(\lambda)\in\operatorname{POL}^{ss}_{d,m\times m}$, with
$\operatorname{rank}Q=2r_{1}$ and $r_{1}<r$, there exists a sequence of skew-
symmetric matrix polynomials
$\\{P^{(k)}(\lambda)\\}_{k\in\mathbb{N}}\subset\operatorname{POL}^{ss}_{d,m\times
m}$ with $\operatorname{rank}P^{(k)}=2r$, for all $k\in\mathbb{N}$, such that
$\lim_{k\rightarrow\infty}P^{(k)}=Q$.

###### Proof.

Let $\mu\in\mathbb{C}$ be such that $\operatorname{rank}Q(\mu)=2r_{1}$, that
is, $\mu$ is not an eigenvalue of $Q(\lambda)$. Since the constant matrix
$Q(\mu)\in\mathbb{C}^{m\times m}$ is skew symmetric, there exist a unitary
matrix $U\in\mathbb{C}^{m\times m}$ and positive real numbers
$s_{1},\ldots,s_{r_{1}}$ such that (see [37, Corollary 2.6.6])

$Q(\mu)=U\left(\begin{bmatrix}0&s_{1}\\\
-s_{1}&0\end{bmatrix}\oplus\dots\oplus\begin{bmatrix}0&s_{r_{1}}\\\
-s_{r_{1}}&0\end{bmatrix}\oplus 0_{m-2r_{1}}\right)U^{T}.$

Then, we define the following constant $m\times m$ skew-symmetric matrix

$E:=U\left(0_{2r_{1}}\oplus\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\oplus\dots\oplus\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\oplus 0_{m-2r}\right)U^{T},$

where there are $r-r_{1}$ blocks equal to $\left[\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right]$. Then, the desired sequence of skew-symmetric
matrix polynomials is defined as $P^{(k)}(\lambda):=Q(\lambda)+\frac{1}{k}E$.
Note that $\operatorname{rank}P^{(k)}=2r$, because
$\operatorname{rank}P^{(k)}\leq\operatorname{rank}Q+\operatorname{rank}(\frac{1}{k}E)=2r_{1}+2r-2r_{1}=2r$
and
$\operatorname{rank}P^{(k)}\geq\operatorname{rank}P^{(k)}(\mu)=\operatorname{rank}(Q(\mu)+\frac{1}{k}E)=2r$.
∎

We recall the following lemma that reveals, for skew-symmetric matrix
polynomials $P$ with odd grade $d$, a relation between
$\overline{\operatorname{O}}({\cal F}_{P})$, where the closure is taken in
$\operatorname{GSYL}^{ss}_{d,m\times m}$, and
$\overline{\operatorname{O}^{c}}({{\cal F}}_{P})$, where the closure is taken
in $\operatorname{PENCIL}^{ss}_{n\times n}$, with $n=md$.

######  Lemma 4.2.

[24, Lemma 5.1] Let $P$ be an $m\times m$ skew-symmetric matrix polynomial
with odd grade $d$ and ${{\cal F}}_{P}$ be its linearization (2). Then
$\overline{\operatorname{O}}({{\cal
F}}_{P})=\overline{\operatorname{O}^{c}}({{\cal
F}}_{P})\cap\operatorname{GSYL}^{ss}_{d,m\times m}$.

We are finally in the position of stating and proving the main result of this
paper.

######  Theorem 4.3.

Let $d,m$, and $r$ be integers such that $d\geq 1$ is even, $m\geq 2$, and
$2\leq 2r\leq(m-1)$. The set of $m\times m$ complex skew-symmetric matrix
polynomials of grade $d$ with rank at most $2r$ is a closed subset of
$\operatorname{POL}_{d,m\times m}^{ss}$ equal to
$\overline{\operatorname{O}}(W)$, where
$W\in\operatorname{POL}^{ss}_{d,m\times m}$ is an $m\times m$ complex skew-
symmetric matrix polynomial of degree exactly $d$ and rank exactly $2r$ with
no elementary divisors at all, with $t$ left minimal indices equal to
$(\beta+1)$ and with $(m-2r-t)$ left minimal indices equal to $\beta$, where
$\beta=\lfloor rd/(m-2r)\rfloor$ and $t\equiv rd\mod(m-2r)$, and with the
right minimal indices equal to the left minimal indices.

###### Proof.

By Lemma 4.1, each skew-symmetric matrix polynomial in
$\operatorname{POL}^{ss}_{d,m\times m}$ of rank $2r_{1}$, with $r_{1}<r$, is
in the closure of the subset of $\operatorname{POL}^{ss}_{d,m\times m}$ formed
by the matrix polynomials of rank exactly $2r$. Therefore it remains to show
that any skew-symmetric matrix polynomial of rank exactly $2r$ in
$\operatorname{POL}^{ss}_{d,m\times m}$ is in the closure of the orbit of $W$
as in the statement.

Denote the complete eigenstructure described in the statement (consisting only
of the left and right minimal indices) by

${\bf
W}:\bigg{\\{}\overbrace{\underbrace{\beta+1,\dots,\beta+1}_{t},\underbrace{\beta,\dots,\beta}_{m-2r-t}}^{\text{left
minimal
indices}},\overbrace{\underbrace{\beta+1,\dots,\beta+1}_{t},\underbrace{\beta,\dots,\beta}_{m-2r-t}}^{\text{right
minimal indices}}\bigg{\\}}.$ (4)

First we show that there exists an $m\times m$ skew-symmetric matrix
polynomial $W\in\operatorname{POL}^{ss}_{d,m\times m}$ of degree exactly $d$
and rank exactly $2r$ that has the complete eigenstructure ${\bf W}$ in (4).
By [22, Thm. 3.3] it is enough to show that the sum of the left (or right)
minimal indices of ${\bf W}$ is equal to $rd$:

$\displaystyle\sum_{1}^{t}(\beta+1)+\sum_{1}^{m-2r-t}\beta=\sum_{1}^{m-2r}\beta+t=(m-2r)\lfloor
rd/(m-2r)\rfloor+t=rd.$

Next, as in Subsection 3.2, we add the term $\lambda^{d+1}\,0$ to $W$ and
denote the (entry-wise identical) result by $\widetilde{W}$, i.e.,
$\widetilde{W}=\lambda^{d+1}\,0+W\in\operatorname{POL}^{ss}_{d+1,m\times m}$,
in order to view the polynomial as a polynomial of odd grade $d+1$. Parts
(a)–(d) in Lemma 3.4 imply that the complete eigenstructure of $\widetilde{W}$
is given by the minimal indices in (4) together with $2r$ infinite elementary
divisors with degree equal to one, i.e., the sequence of partial
multiplicities at $\infty$ of $\widetilde{W}$ is
$\gamma_{1}=\gamma_{1}=\gamma_{2}=\gamma_{2}=\dots=\gamma_{r}=\gamma_{r}=1$.

For every skew-symmetric $m\times m$ matrix polynomial $P$ of odd grade $d+1$
and rank at most $2r$, the linearization ${\cal F}_{P}$ in (2) has rank equal
to $\operatorname{rank}(P)+md$, i.e., rank at most $2r+md$, because ${\cal
F}_{P}$ is unimodularly equivalent to $P\oplus I_{md}$. The linearization
${{\cal F}}_{\widetilde{W}}$ of the matrix polynomial $\widetilde{W}$ is an
$m(d+1)\times m(d+1)$ skew-symmetric matrix pencil with rank $2r+md$ and, by
Theorem 3.5, the KCF of ${{\cal F}}_{\widetilde{W}}$ is the direct sum of the
following blocks:

$\begin{split}\Big{\\{}\underbrace{{\cal L}_{\beta+\eta+1},\dots,{\cal
L}_{\beta+\eta+1}}_{t},\underbrace{{\cal L}_{\beta+\eta},\dots,{\cal
L}_{\beta+\eta}}_{m-2r-t},\underbrace{{\cal L}_{\beta+\eta+1}^{T},\dots,{\cal
L}_{\beta+\eta+1}^{T}}_{t},\underbrace{{\cal L}_{\beta+\eta}^{T},\dots,{\cal
L}_{\beta+\eta}^{T}}_{m-2r-t},\\\ \underbrace{{\cal E}_{1}(\infty),\dots,{\cal
E}_{1}(\infty)}_{2r}\Big{\\}},\end{split}$ (5)

where $\eta=\frac{1}{2}d$. Next, we show that the KCF of ${\cal
F}_{\widetilde{W}}$ as in (5) coincides with the KCF of the most generic skew-
symmetric matrix pencil ${\cal W}$ of rank $2w=2r+md$ and size $n\times n$,
where $n=m(d+1)$, having exactly $r$ ${\cal K}$-blocks associated to the
infinite eigenvalue in its skew-symmetric KCF. This generic KCF is given in
Theorem 2.4, namely:

$\begin{split}\Big{\\{}\underbrace{{\cal L}_{\alpha+1},\dots,{\cal
L}_{\alpha+1}}_{s},\underbrace{{\cal L}_{\alpha},\dots,{\cal
L}_{\alpha}}_{n-2w-s},\underbrace{{\cal L}_{\alpha+1}^{T},\dots,{\cal
L}_{\alpha+1}^{T}}_{s},\underbrace{{\cal L}_{\alpha}^{T},\dots,{\cal
L}_{\alpha}^{T}}_{n-2w-s},\underbrace{{\cal E}_{1}(\infty),\dots,{\cal
E}_{1}(\infty)}_{2r}\Big{\\}}.\end{split}$ (6)

In other words, we are going to show that the numbers and the sizes of the
${\cal L}$ and ${\cal L}^{T}$ blocks in (5) and (6) coincide, namely
$\beta+d/2=\alpha$, $s=t$, and $m-2r-t=n-2w-s$.

For the sizes of the blocks we have

$\displaystyle\beta+\frac{d}{2}$
$\displaystyle=\left\lfloor\frac{rd}{m-2r}\right\rfloor+\frac{d}{2}=\left\lfloor\frac{(m-2r)d+2rd}{2(m-2r)}\right\rfloor=\left\lfloor\frac{md}{2(m-2r)}\right\rfloor$
$\displaystyle=\left\lfloor\frac{md+2r-2r}{2(m(d+1)-(md+2r))}\right\rfloor=\left\lfloor\frac{2w-2r}{2(n-2w)}\right\rfloor=\alpha.$

For the numbers of the blocks, we have

$\displaystyle t$ $\displaystyle\equiv rd\equiv(m-2r)d/2+rd\mod(m-2r)$
$\displaystyle\equiv(md+2r-2r)/2\mod(m(d+1)-(md+2r))$ $\displaystyle\equiv
w-r\equiv s\mod(n-2w)$

and, since $s,t<n-2w$, it must be $t=s$. Moreover,

$m-2r-t=m(d+1)-md-2r-t=n-2w-s.$

Thus ${{\cal F}}_{\widetilde{W}}$ is congruent to the most generic skew-
symmetric matrix pencil $\mathcal{W}$, with exactly $r$ ${\cal K}$-blocks
corresponding to the infinite eigenvalue in its skew-symmetric KCF and of rank
$2w=2r+md$, obtained in Theorem 2.4, since they both are skew-symmetric and
have the same KCF. Therefore $\operatorname{O}^{c}({{\cal
F}}_{\widetilde{W}})=\operatorname{O}^{c}(\mathcal{W})$.

Given any $P\in\operatorname{POL}^{ss}_{d,m\times m}$ of rank exactly $2r$,
the entry-wise identical matrix polynomial
$\widetilde{P}=\lambda^{d+1}\,0+P\in\operatorname{POL}^{ss}_{d+1,m\times m}$
has rank exactly $2r$ and exactly $2r$ infinite elementary divisors, by Lemma
3.4-(a)-(d). So, by Theorem 2.4, we have that
$\overline{\operatorname{O}^{c}}(\mathcal{W})\supseteq\operatorname{O}^{c}({{\cal
F}}_{\widetilde{P}})$, thus $\overline{\operatorname{O}^{c}}({{\cal
F}}_{\widetilde{W}}){=\overline{\operatorname{O}^{c}}({\cal
W})}\supseteq\operatorname{O}^{c}({{\cal F}}_{\widetilde{P}})$. Therefore
$\overline{\operatorname{O}^{c}}({{\cal
F}}_{\widetilde{W}})\cap\operatorname{GSYL}^{ss}_{d+1,m\times
m}\supseteq\operatorname{O}^{c}({{\cal
F}}_{\widetilde{P}})\cap\operatorname{GSYL}^{ss}_{d+1,m\times m}$, and Lemma
4.2 implies $\overline{\operatorname{O}}({{\cal
F}}_{\widetilde{W}})\supseteq\operatorname{O}({{\cal F}}_{\widetilde{P}})$.
According to the discussion after (3), $\overline{\operatorname{O}}({{\cal
F}}_{\widetilde{W}})\supseteq\operatorname{O}({{\cal F}}_{\widetilde{P}})$ is
equivalent to
$\overline{\operatorname{O}}(\widetilde{W})\supseteq\operatorname{O}(\widetilde{P})$
and by Lemma 3.4-(g)-3 the latter is equivalent to
$\overline{\operatorname{O}}(W)\supseteq\operatorname{O}(P)$. Therefore any
$m\times m$ skew-symmetric matrix polynomial of grade $d$, with rank exactly
$2r$ belongs to $\overline{\operatorname{O}}(W)$. This completes the proof (we
remind that the skew-symmetric matrix polynomials of grade $d$ with rank
smaller than $2r$ are treated at the beginning of this proof). ∎

######  Example 4.4.

In Table 2 we provide examples of the eigenstructures (namely, the left
minimal indices) of generic skew-symmetric matrix polynomials of grade $2$
with various sizes (from $3\times 3$ to $7\times 7$) and ranks ($2,4$, and
$6$) obtained from Theorem 4.3.

| $3\times 3$ | $4\times 4$ | $5\times 5$ | $6\times 6$ | $7\times 7$
---|---|---|---|---|---
rank 2 |  {2} |  {1, 1} |  {1, 1, 0} |  {1, 1, 0, 0} |  {1, 1, 0, 0, 0}
rank 4 | - | - |  {4} |  {2, 2} |  {2, 1, 1}
rank 6 | - | - | - | - |  {6}
Table 2:  Left minimal indices of the generic skew-symmetric matrix polynomial
of grade 2 and of certain sizes (columns correspond to different sizes) and
rank (rows correspond to different ranks). Note that the right minimal indices
are equal to the left minimal indices, and no elementary divisors are present
in these polynomials.

Note that Theorem 4.3 is an even-grade-version of the corresponding theorem
for the odd grade skew-symmetric matrix polynomials [24, Thm. 5.2]. This
allows us to state the following theorem that does not depend on the parity of
$d$.

######  Theorem 4.5.

Let $m,r$, and $d$ be integers such that $m\geq 2$, $d\geq 1$, and $2\leq
2r\leq(m-1)$. The set of $m\times m$ complex skew-symmetric matrix polynomials
of grade $d$ with rank at most $2r$ is a closed subset of
$\operatorname{POL}_{d,m\times m}^{ss}$ equal to
$\overline{\operatorname{O}}(W)$, where
$W\in\operatorname{POL}^{ss}_{d,m\times m}$ is an $m\times m$ complex skew-
symmetric matrix polynomial of degree exactly $d$ and rank exactly $2r$ with
no elementary divisors at all, with $t$ left minimal indices equal to
$(\beta+1)$ and with $(m-2r-t)$ left minimal indices equal to $\beta$, where
$\beta=\lfloor rd/(m-2r)\rfloor$ and $t=rd\mod(m-2r)$, and with the right
minimal indices equal to the left minimal indices.

###### Proof.

If $d$ is odd then the statement coincides with [24, Thm. 5.2], and if $d$ is
even then it coincides with Theorem 4.3. ∎

##  5 Codimension computations

We start by recalling that the congruence orbit of an $n\times n$ skew-
symmetric pencil $\lambda A-B$ is a differential manifold in the complex
$n^{2}-n$ dimensional space of skew-symmetric pencils
$\operatorname{PENCIL}^{ss}_{n\times n}$. For $\lambda A-B$, define the
dimension of $\operatorname{O}^{c}(\lambda A-B)$ to be the dimension of the
tangent space to this orbit at the point $\lambda A-B$, namely:

$\operatorname{T}_{\lambda
A-B}^{c}:=\\{\lambda(X^{T}A+AX)-(X^{T}B+BX):X\in{\mathbb{C}}^{n\times n}\\}$

The orthogonal complement (with respect to the Frobenius inner product) to
$\operatorname{T}_{\lambda A-B}^{c}$, is called the normal space to
$\operatorname{O}^{c}(\lambda A-B)$ at the point $\lambda A-B$. The dimension
of the normal space is the codimension of the congruence orbit of $\lambda
A-B$ and is equal to $n(n-1)$ minus the dimension of the congruence orbit of
$\lambda A-B$. Explicit expressions for the codimensions of congruence orbits
of skew-symmetric pencils in $\operatorname{PENCIL}_{n\times n}^{ss}$ are
presented in [29], see also [21], and implemented in the MCS (Matrix Canonical
Structure) Toolbox [25]. Recall that, for $Q\in\operatorname{POL}_{d,m\times
m}^{ss}$ of odd grade, we define the codimension of
$\operatorname{O}(Q)\subset\operatorname{POL}_{d,m\times m}^{ss}$ to be
$\operatorname{cod}_{\operatorname{POL}}\operatorname{O}(Q):=\operatorname{cod}_{\operatorname{GSYL}}\operatorname{O}({{\cal
F}}_{Q})$, where by $\operatorname{cod}_{X}(\operatorname{O})$ we mean the
codimension of the orbit $\operatorname{O}$ in the space $X$, see [22, 24].
Moreover, $\operatorname{cod}_{\operatorname{GSYL}}\operatorname{O}({{\cal
F}}_{P})=\operatorname{cod}_{\operatorname{PENCIL}}\operatorname{O}^{c}({{\cal
F}}_{P})$, by [22, Sect. 6].

Now let $P\in\operatorname{POL}_{d,m\times m}^{ss}$ be a matrix polynomial of
even grade $d$ and
$\widetilde{P}:=0\,\lambda^{d+1}+P\in\operatorname{POL}_{d+1,m\times m}^{ss}$
be a matrix polynomial of odd grade $d+1$. Denote by
$\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$ the space of the linearizations
(2) of skew-symmetric polynomials $\widetilde{P}$ of odd grade $d+1$ with zero
coefficient for $\lambda^{d+1}$. For a polynomial $P$ of even grade $d$, we
define the codimension of the orbit,
$\operatorname{cod}_{\operatorname{POL}}\operatorname{O}(P)$, as
$\operatorname{cod}_{\operatorname{GSYL}^{0}}\operatorname{O}({\cal
F}_{\widetilde{P}})$, i.e., the codimension of $\operatorname{O}({\cal
F}_{\widetilde{P}})$ in the space $\operatorname{GSYL}_{d+1,m\times
m}^{ss,0}$. Recall also that the dimension of $\operatorname{O}({\cal
F}_{\widetilde{P}})$ in the space $\operatorname{GSYL}_{d+1,m\times m}^{ss}$
is defined as the dimension of $\operatorname{T}_{{\cal
F}_{\widetilde{P}}}^{c}\cap\operatorname{GSYL}_{d+1,m\times m}^{ss}$ and,
similarly, the dimension of $\operatorname{T}_{{\cal
F}_{\widetilde{P}}}^{c}\cap\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$ is
equal to the dimension of $\operatorname{O}({\cal F}_{\widetilde{P}})$ in
$\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$. In Lemma 5.1 we show that these
two dimensions are equal to each other.

######  Lemma 5.1.

Let
$\widetilde{P}=0\,\lambda^{d+1}+P_{d}\lambda^{d}+\dots+P_{1}\lambda+P_{0}\,{\in\operatorname{POL}^{ss}_{d+1,m\times
m}}$ be a skew-symmetric matrix polynomial. The dimension of
$\operatorname{O}({{\cal F}}_{\widetilde{P}})$ in
$\operatorname{GSYL}_{d+1,m\times m}^{ss}$is equal to the dimension of
$\operatorname{O}({{\cal F}}_{\widetilde{P}})$ in
$\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$.

###### Proof.

The tangent space $\operatorname{T}_{{\cal
F}_{\widetilde{P}}}^{c}={\\{X^{T}{\cal F}_{\widetilde{P}}+{\cal
F}_{\widetilde{P}}X:X\in{\mathbb{C}}^{n\times n}\\}}$ to
$\operatorname{O}^{c}({{\cal F}}_{\widetilde{P}})$ at the point ${\cal
F}_{\widetilde{P}}$ consist of the skew-symmetric pencils of the following
shape:

${X^{T}{\cal F}_{\widetilde{P}}+{\cal
F}_{\widetilde{P}}X}=\lambda\begin{bmatrix}0_{m\times m}&*&\dots&*\\\
*&*&\dots&*\\\ \vdots&\vdots&&\vdots\\\ *&*&\dots&*\\\
\end{bmatrix}-\begin{bmatrix}*&*&\dots&*\\\ *&*&\dots&*\\\
\vdots&\vdots&&\vdots\\\ *&*&\dots&*\\\ \end{bmatrix},$ (7)

where $*$ denote possibly non-zero entries. The zero block in the $(1,1)$
entry of the $\lambda$-matrix of (7) ensures that $\operatorname{T}_{{\cal
F}_{\widetilde{P}}}\cap\operatorname{GSYL}_{d+1,m\times m}^{ss}$ is equal to
$\operatorname{T}_{{\cal
F}_{\widetilde{P}}}\cap\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$, and so
their dimensions are equal to each other. ∎

Recall that the dimension and codimension of any given orbit sum up to the
dimension of the whole space. Note also that
$\dim\operatorname{GSYL}_{d+1,m\times
m}^{ss}=\dim\operatorname{GSYL}_{d+1,m\times m}^{ss,0}+\frac{m(m-1)}{2}$.
Thus, taking into account Lemma 5.1, the codimension of
$\operatorname{O}({{\cal F}}_{\widetilde{P}})$ in
$\operatorname{GSYL}_{d+1,m\times m}^{ss}$ has to be $\frac{m(m-1)}{2}$ larger
than the codimension of $\operatorname{O}({{\cal F}}_{\widetilde{P}})$ in
$\operatorname{GSYL}_{d+1,m\times m}^{ss,0}$.

Therefore, for the generic $m\times m$ skew-symmetric matrix polynomial $P$
with even grade $d$ identified in Theorem 4.3, we have

$\displaystyle\operatorname{cod}_{\operatorname{POL}}\operatorname{O}(P)$
$\displaystyle=\operatorname{cod}_{\operatorname{GSYL}^{0}}\operatorname{O}({{\cal
F}}_{\widetilde{P}})=\operatorname{cod}_{\operatorname{GSYL}}\operatorname{O}({{\cal
F}}_{\widetilde{P}})-\frac{m(m-1)}{2}$
$\displaystyle=\operatorname{cod}_{\operatorname{PENCIL}}\operatorname{O}^{c}({\cal
F}_{\widetilde{P}})-\frac{m(m-1)}{2}$
$\displaystyle=\operatorname{cod}_{\operatorname{PENCIL}}\operatorname{O}^{c}({\cal
W})-\frac{m(m-1)}{2},$ (8)

where ${\cal W}$ is the pencil in Theorem 2.4 with the identifications
$n=m(d+1)$ and $w=(md+2r)/2$. Using [29, Thm. 3], see also [25, Thm. 2.8],
applied to the skew-symmetric pencil ${\cal W}$ in (1), the codimension of
$\operatorname{O}^{c}({\cal W})$ in $\operatorname{PENCIL}_{n\times n}^{ss}$
is a sum of the part corresponding to the ${\cal K}$ blocks, namely $r(2r-1)$,
the part corresponding to the interaction between the ${\cal K}$ blocks and
${\cal M}$ blocks, namely $2r(n-2w)$, and the part corresponding to the ${\cal
M}$ blocks, namely $\sum_{i<j}(2\max\\{m_{i},m_{j}\\}+\varepsilon_{ij})$,
where $m_{i}$ and $m_{j}$ are the indices of the ${\cal M}$ blocks (either
$\alpha$ or $\alpha+1$), $\varepsilon_{ij}=2$ if $m_{i}=m_{j}$ and
$\varepsilon_{ij}=1$ otherwise, resulting in:

$\displaystyle{\operatorname{cod}_{\operatorname{PENCIL}}}\operatorname{O}^{c}({\cal
W})=(n-2w-s)(n-2w-s-1)(2\alpha+2)/2$
$\displaystyle\phantom{a}+s(s-1)(2(\alpha+1)+2)/2+s(n-2w-s)(2(\alpha+1)+1)+2r(n-2w)+r(2r-1)$
$\displaystyle\phantom{a}=(n-2w-s)(n-2w-s-1)(\alpha+1)+s(s-1)(\alpha+1)+s(s-1)$
$\displaystyle\phantom{a}+2s(n-2w-s)(\alpha+1)+s(n-2w-s)+2r(n-2w)+r(2r-1)$
$\displaystyle\phantom{a}=(\alpha+1)((n-2w-s)(n-2w-1)+s(n-2w-1))$
$\displaystyle\phantom{a}+s(n-2w-1)+2r(n-2w)+r(2r-1)$
$\displaystyle\phantom{a}=(n-2w)(n-2w-1)(\alpha+1)+s(n-2w-1)+2r(n-2w)+r(2r-1)$
$\displaystyle\phantom{a}=(n-2w-1)((n-2w)\alpha+s+n-2w)+2r(n-2w)+r(2r-1)$
$\displaystyle\phantom{a}=(n-2w-1)(n-w-r)+2r(n-2w)+r(2r-1).$

Substituting $n=m(d+1)$ and $w=(md+2r)/2$ in the previous identity we obtain:

$\displaystyle\operatorname{cod}_{\operatorname{GSYL}}\operatorname{O}({\cal
F}_{\widetilde{P}})$ $\displaystyle=(n-2w-1)(n-w-r)+2r(n-2w)+r(2r-1)$
$\displaystyle=(m(d+1)-2r-md-1)(m(d+1)-(2r+md)/2-r)$
$\displaystyle+2r(m(d+1)-2r-md)+r(2r-1)$
$\displaystyle=(m-2r-1)(md+2m-4r)/2+2mr-{2r^{2}}-r$
$\displaystyle=(m-2r-1)(md+m-2r)/2+m(m-1)/2.$

Taking into account equation (8), we obtain:

$\displaystyle{\operatorname{cod}_{\operatorname{POL}}\operatorname{O}(P)}$
$\displaystyle=\operatorname{cod}_{\operatorname{GSYL}^{0}}\operatorname{O}({\cal
F}_{\widetilde{P}})=\operatorname{cod}_{\operatorname{GSYL}}\operatorname{O}({\cal
F}_{\widetilde{P}})-m(m-1)/2$ $\displaystyle=(m-2r-1)(md+m-2r)/2,$

which coincides with the codimension found in [24, p. 15] for the odd-grade
case.

##  6 Conclusions and future work

This paper closes the knowledge gap for skew-symmetric matrix polynomials with
bounded rank and fixed grade. Namely, it provides the generic complete
eigenstructure for skew-symmetric matrix polynomials of rank at most $2r$ and
even grade $d$. It is also the first paper that tackles such problem for the
structured matrix polynomials of even grade. Moreover, the obtained formulas
(for the generic eigenstructures and codimensions) are exactly the same as in
the case of odd $d$ obtained in [24], showing that the restriction to the odd
case was motivated by a need of using the linearization techniques.

The extensions of this result to another classes of structured matrix
polynomials of even grade, such as symmetric, symmetric/skew-symmetric, or
palindromic, is a part of our future work.

## Acknowledgements

The work of A. Dmytryshyn was supported by the Swedish Research Council (VR)
grant 2021-05393. The work of F. De Terán and F. M. Dopico has been partially
funded by the Agencia Estatal de Investigación of Spain through grants
PID2019-106362GB-I00 MCIN/ AEI/ 10.13039/ 501100011033/ and RED2022-134176-T,
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).

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