# On matrices in finite free position

Octavio Arizmendi<EMAIL_ADDRESS>,  Franz Lehner<EMAIL_ADDRESS>graz.ac.at and  Amnon Rosenmann<EMAIL_ADDRESS>
###### Abstract.

We study pairs $(A,B)$ of square matrices that are in additive (resp.
multiplicative) finite free position, that is, the characteristic polynomial
$\chi_{A+B}(x)$ (resp. $\chi_{AB}(x)$) equals the additive finite free
convolution $\chi_{A}(x)\boxplus\chi_{B}(x)$ (resp. the multiplicative finite
free convolution $\chi_{A}(x)\boxtimes\chi_{B}(x)$), which equals the expected
characteristic polynomial $\mathbb{E}_{U}\,[\chi_{A+U^{*}BU}(x)]$ (resp.
$\mathbb{E}_{U}\,[\chi_{AU^{*}BU}(x)]$) over the set of unitary matrices $U$.
We examine the lattice of algebraic varieties of matrices consisting of finite
free complementary pairs with respect to the additive (resp. multiplicative)
convolution. We show that these pairs include the diagonal matrices vs. the
principally balanced matrices, the upper (lower) triangular matrices vs. the
upper (lower) triangular matrices with a single eigenvalue, and the scalar
matrices vs. the set of all square matrices.

###### Key words and phrases:

Random matrices, free probability, polynomial convolutions, principal minors

###### 2010 Mathematics Subject Classification:

60B20, 46L54

Co-funded by the European Union F.L. was partly supported by the H2020-MSCA-
RISE project 734922 - CONNECT

A.R. was supported by the Austrian Science Fund (FWF) project p29355

###### Contents

  1. 1 Introduction
  2. 2 Matrices in additive finite free position
  3. 3 Matrices of dimension $2\times 2$
  4. 4 The lattice of additive finite free varieties
  5. 5 Additive finite free complementary pairs
    1. 5.1 Diagonal, principally balanced and symmetric matrices
    2. 5.2 Scalar matrices
    3. 5.3 Triangular matrices
    4. 5.4 Conclusion
  6. 6 Moments of sums of matrices that are in additive FFP
  7. 7 Matrices in multiplicative finite free position
    1. 7.1 Matrices of dimension $2\times 2$
    2. 7.2 Matrices of dimension $3\times 3$
    3. 7.3 The lattice of multiplicative finite free varieties
    4. 7.4 Diagonal, principally balanced and symmetric matrices
    5. 7.5 Triangular matrices
    6. 7.6 Scalar matrices
    7. 7.7 Matrices in multiplicative FFP with themselves
    8. 7.8 Moments of products of matrices that are in multiplicative FFP

##  1\. Introduction

Finite free probability was introduced by Marcus, Spielman and Srivastava [10]
and Marcus [9] in connection with the solution of the Kadison-Singer
conjecture. They introduced additive and multiplicative convolutions of
characteristic polynomials of matrices over the real and complex numbers. In
fact, these convolutions were already studied in the 1920s by Walsh [18] and
Szegő [16], who proved among other results that under certain conditions they
preserve real rootedness. In [10] explicit formulas for the convolutions were
introduced, showing that they express the expectations of the characteristic
polynomials of unitarily (orthogonally) invariant random matrices, thus
linking them to free probability. That is, when $A$ and $B$ are normal
(symmetric) matrices then the additive convolution of their characteristic
polynomials equals the expectation of the characteristic polynomials of
$A+U^{*}BU$, where the expectation is over randomly distributed unitary
(orthogonal) matrices $U$. A similar result holds for normal matrices with
respect to multiplicative convolution.

The analogues of these convolutions in the setting of tropical algebra were
studied in [12].

In [9], Marcus developed further the findings of [10] to reach a new theory of
“finite free probability”, a finite version of Free Probability that was
introduced by Voiculescu [17], with many analogous properties. The approach
used by Marcus was mainly analytic, through several transforms that are the
analogues of the transforms applied in free probability. An alternative
combinatorial approach via set partitions was chosen by Arizmendi et al. [2,
1] who established an analogue of Speicher’s combinatorial approach to free
probability [14].

In the present paper we study pairs of matrices $A,B$ which are in finite free
position (FFP), that is, such that the characteristic polynomial of the sum
satisfies the identity $\chi_{A+B}(x)=\chi_{A}\boxplus\chi_{B}$, the additive
finite free convolution of $\chi_{A}$ and $\chi_{B}$, which further equals the
expected characteristic polynomial $\mathbb{E}_{U}\,[\chi_{A+U^{*}BU}(x)]$
over the set of unitary matrices $U$. We consider the analogous question in
the multiplicative case.

The paper is organized as follows.

In Section 2 we recall the definition of additive finite free convolution and
FFP.

In Section 3 we study in detail the simplest case of $2\times 2$ matrices in
additive FFP.

In Section 4 we change the point of view from single matrices to the lattice
of varieties of matrices $\mathcal{V}$ and $\mathcal{W}$, which are maximal
with respect to the property that each matrix in $\mathcal{V}$ is in additive
FFP with each matrix in $\mathcal{W}$. We say that such families form a finite
free complementary pair. We give an upper bound for the rank of each such
variety.

In Section 5 we focus on specific complementary pairs: diagonal and
principally balanced matrices (matrices with the property that for each $i$
the values of all principal minors of size $i$ coincide); upper triangular and
upper triangular matrices with constant diagonal; scalar matrices and the set
of all square matrices.

In Section 6, we examine moments and cumulants of matrices $A$ and $B$ that
are in additive FFP and show that the moments of $A+B$ can be expressed in
terms of the moments of $A$ and $B$, and compute explicit formulas for the
first four moments. We also show that a matrix $A$ that is in additive FFP
with itself must have a single eigenvalue.

In Section 7 we extend our analysis to matrices the multiplicative FFP. In
particular, we show that the three complementary pairs of varieties mentioned
above form complementary pairs also in the multiplicative case. In general,
however, the pairs of matrices that are in additive FFP and those in
multiplicative FFP are not the same.

##  2\. Matrices in additive finite free position

Additive convolution of polynomials can be defined as follows [10].

######  Definition 2.1.

Let $p(x)=\sum_{k=0}^{n}a_{k}x^{n-k}$, $q(x)=\sum_{k=0}^{n}b_{k}x^{n-k}$ be
two polynomials of degree $n$ over $\mathbb{C}$. The additive convolution of
$p(x)$ and $q(x)$, denoted $p(x)\boxplus q(x)$, is

$\displaystyle p(x)\boxplus q(x)$
$\displaystyle:=\sum_{k=0}^{n}\left(\sum_{i+j=k}\frac{{n-i\choose
j}}{{n\choose j}}a_{i}b_{j}\right)x^{n-k}$ (2.1)
$\displaystyle=\frac{1}{n!}\sum_{k=0}^{n}\frac{1}{(n-k)!}\left(\sum_{i+j=k}(n-i)!(n-j)!a_{i}b_{j}\right)x^{n-k}$
$\displaystyle=\frac{1}{n!}\sum_{k=0}^{n}p^{(k)}(x)q^{(n-k)}(0)=\frac{1}{n!}\sum_{k=0}^{n}p^{(k)}(0)q^{(n-k)}(x),$

where we denote by $p^{(k)}$ the $k$-th derivative of $p$ and $q^{(n-k)}$ is
the $(n-k)$-th derivative of $q$.

######  Example 2.2.

The neutral element with respect to the operation of additive convolution on
polynomials of degree $n$ is $x^{n}$: when $p(x)$ is of degree $n$ and
$q(x)=x^{n}$ then

$p(x)\boxplus
q(x)=\frac{1}{n!}\sum_{k=0}^{n}p^{(k)}(x)q^{(n-k)}(0)=\frac{1}{n!}p^{(0)}(x)q^{(n)}(0)=\frac{p(x)n!}{n!}=p(x),$

and similarly, $x^{n}\boxplus p(x)=p(x)$.

We denote by $\mathcal{M}_{n}$ the set of $n\times n$ matrices over
$\mathbb{C}$. Given a matrix $A\in\mathcal{M}_{n}$, its characteristic
polynomial is $\chi_{A}(x)=\det(xI-A)$.

Following is one of the main results of [10]. Recall that a signed permutation
matrix is a square matrix with each row and each column having exactly one
non-zero entry which is either $1$ or $-1$.

######  Theorem 2.3.

[10] Let $A,B\in\mathcal{M}_{n}$ be normal ($A,B\in\mathbb{R}^{n\times n}$ be
symmetric) matrices. Then

(2.2)
$\chi_{A}(x)\boxplus\chi_{B}(x)=\int_{\mathcal{U}(n)}\chi_{A+U^{*}BU}(x)\,dU=\frac{1}{2^{n}n!}\sum_{P\in\mathcal{P}^{\pm}(n)}\chi_{A+P^{T}BP}(x),$

where the expectation is taken over the set of unitary (orthogonal) matrices
$\mathcal{U}(n)$ or the signed permutation matrices $\mathcal{P}^{\pm}(n)$.

In the present paper we study pairs of matrices $A$ and $B$ which satisfy
equality (2.2) without taking the expected values of conjugations of $B$.

######  Definition 2.4.

The matrices $A,B\in\mathcal{M}_{n}$ are in additive finite free position (or
in additive FFP), if

$\chi_{A+B}(x)=\chi_{A}(x)\boxplus\chi_{B}(x).$

Two families $\mathcal{A},\mathcal{B}\subseteq\mathcal{M}_{n}$ are in additive
finite free position (in additive FFP) if
$\chi_{A+B}(x)=\chi_{A}(x)\boxplus\chi_{B}(x)$ for every $A\in\mathcal{A}$ and
$B\in\mathcal{B}$.

######  Remarks 2.5.

  1.   2. (1)

As shown in [9], the property of being in additive FFP can be expressed in
terms of mixed discriminants [3].

  3. (2)

We notice that $\chi_{A}(x)\boxplus\chi_{B}(x)$ only depends on the
characteristic polynomials of the matrices and not on the matrices themselves.

  4. (3)

This is in general not the case for $\chi_{A+B}(x)$, we can only assert that
for any pair of matrices $A,B\in\mathcal{M}_{n}$, the two leading monomials of
$\chi_{A+B}(x)$ and $\chi_{A}(x)\boxplus\chi_{B}(x)$ coincide and are equal to
$x^{n}$ and $-\operatorname{\mathrm{Tr}}(A+B)x^{n-1}$, respectively.

  5. (4)

If $A$ and $B$ are in additive FFP then so are $PAP^{-1}$ and $PBP^{-1}$, for
any regular matrix $P$.

##  3\. Matrices of dimension $2\times 2$

Let us start with an explicit analysis of $2\times 2$ matrices.

######  Proposition 3.1.

The matrices $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{{2}\times{2}}$ are in
additive FFP if and only if
$(a_{11}-a_{22})(b_{22}-b_{11})=2(a_{12}b_{21}+a_{21}b_{12})$.

###### Proof.

Let

$A=\begin{bmatrix}[r]a_{11}&a_{12}\\\
a_{21}&a_{22}\end{bmatrix}\quad\mbox{and}\quad
B=\begin{bmatrix}[r]b_{11}&b_{12}\\\ b_{21}&b_{22}\end{bmatrix}.$

Then

$\displaystyle\chi_{A+B}(x)$
$\displaystyle=x^{2}-(a_{11}+a_{22}+b_{11}+b_{22})x$
$\displaystyle+(a_{11}+b_{11})(a_{22}+b_{22})-(a_{12}+b_{12})(a_{21}+b_{21})$

and

$\displaystyle\chi_{A}(x)$
$\displaystyle\boxplus\chi_{B}(x)=x^{2}-(a_{11}+a_{22}+b_{11}+b_{22})x$
$\displaystyle+(a_{11}a_{22}-a_{12}a_{21}+b_{11}b_{22}-b_{12}b_{21})+\frac{1}{2}(a_{11}+a_{22})(b_{11}+b_{22}).$

We see that both $\chi_{A}(x)\boxplus\chi_{B}(x)$ and
$\chi_{A}(x)\boxplus\chi_{B}(x)$ are monic polynomials with the same
coefficient of $x$: negative the sum of the traces (in fact, this holds for
any two $n\times n$ matrices, cf. Remark 2.5). By comparing the free terms of
both polynomials we get that $A$ and $B$ are in additive FFP if and only if

(3.1) $(a_{11}-a_{22})(b_{22}-b_{11})=2(a_{12}b_{21}+a_{21}b_{12}).$

∎

Since the property of being in additive FFP is invariant under conjugating
both $A$ and $B$ with the same matrix, we can assume that $A$ is of the form

$A=\begin{bmatrix}[c]a_{11}&a_{12}\\\ 0&a_{22}\end{bmatrix}.$

Then, condition (3.1) for $A$ and $B$ to be in additive FFP becomes:

(3.2) $(a_{11}-a_{22})(b_{22}-b_{11})=2a_{12}b_{21}.$

By (3.2), $A$ is in additive FFP with itself if and only if

$a_{11}=a_{22},$

that is, A has a single eigenvalue. In fact, by Theorem 6.3, in general, an
$n\times n$ complex matrix is in additive FFP with itself if and only if it
has a single eigenvalue.

When $A$ (or $B$) is a scalar matrix then both sides of (3.1) are zero, so
that $A$ and $B$ are in additive FFP. Moreover, when $A$ and $B$ are in
additive FFP then it is easy to see that equality (3.1) holds also for
$\lambda_{1}A+\lambda_{2}I$ and $\mu_{1}B+\mu_{2}I$, for every
$\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}\in\mathbb{C}$. In fact, we have the
following more general result. We will see later that it does not hold in
higher dimensions, see Remark 4.4.

######  Proposition 3.2.

Let $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{{2}\times{2}}$ be in additive FFP.
Then $p(A)$ and $q(B)$ are in additive FFP, for any polynomials $p(x)$ and
$q(x)$. Moreover, if $A$ is regular then $A^{-1}$ and $q(B)$ are in additive
FFP.

###### Proof.

We infer from the Cayley-Hamilton theorem that every holomorphic function
$f(A)$ can be represented as a linear function of the form $a_{0}I+a_{1}A$ and
the claim follows from Proposition 4.3. ∎

##  4\. The lattice of additive finite free varieties

In this section we extend the discussion from single matrices to sets of
matrices in FFP. To this end we consider the set
$\mathcal{V}\subseteq\mathcal{M}_{n}$ of all matrices that are in additive FFP
with every matrix $A$ in a given set $\mathcal{A}$. In fact this set
$\mathcal{V}$ forms an algebraic variety (algebraic set).

For general $n\times n$ matrices $A=(A_{ij})$ and $B=(B_{ij})$, let

$\chi_{A+B}(x)=\sum_{k=0}^{n}c_{k}(A_{11},A_{12},\ldots,A_{nn},B_{11},B_{12},\ldots,B_{nn})x^{n-k}$

and

$\chi_{A}(x)\boxplus\chi_{B}(x)=\sum_{k=0}^{n}d_{k}(A_{11},A_{12},\ldots,A_{nn},B_{11},B_{12},\ldots,B_{nn})x^{n-k},$

where $c_{k}(A_{11},\ldots,A_{nn},B_{11},\ldots,B_{nn})$,
$d_{k}(A_{11},\ldots,A_{nn},B_{11},\ldots,B_{nn})$ are multilinear homogeneous
polynomials in the variables $A_{11},\ldots,A_{nn}$, $B_{11},\ldots,B_{nn}$ of
degree $k$. By definition 2.1 of the additive convolution, it is clear that a
coefficient of $x^{n-k}$ in $\chi_{A}(x)\boxplus\chi_{B}(x)$ has as summands
the coefficient $a_{k}$ of $\chi_{A}(x)$, as well as the coefficient $b_{k}$
of $\chi_{B}(x)$. These are also summands of $\chi_{A+B}(x)$. Moreover, the
coefficients of of $x^{n}$ and of $x^{n-1}$ (1 and
$-\operatorname{\mathrm{Tr}}(A+B)$, respectively) are the same in
$\chi_{A}(x)\boxplus\chi_{B}(x)$ and in $\chi_{A+B}(x)$, That is, two specific
matrices $A=(a_{ij}),B=(b_{ij})\in\mathcal{M}_{n}$ are in additive FFP if and
only if

$c_{k}(a_{11},\ldots,a_{nn},b_{11},\ldots,b_{nn})=d_{k}(a_{11},\ldots,a_{nn},b_{11},\ldots,b_{nn}),$

for $k=2,\ldots,n$.

Suppose now that $A=(a_{ij})\in\mathcal{M}_{n}$ is a specific matrix. Then the
matrices $B$ that are in additive FFP with $A$ are the zero locus of the
following set of polynomials over $\mathbb{C}$:

$\displaystyle p_{A,k}$ $\displaystyle(B_{11},\ldots,B_{nn}):=$ $\displaystyle
c_{k}(a_{11},\ldots,a_{nn},B_{11},\ldots,B_{nn})-d_{k}(a_{11},\ldots,a_{nn},B_{11},\ldots,B_{nn}),$

for $k=2,\ldots,n$. We associate with $A$ the ideal $I_{A}$ generated by these
polynomials:

$I_{A}:=\langle\,p_{A,2}(B_{11},\ldots,B_{nn}),\ldots,p_{A,n}(B_{11},\ldots,B_{nn})\,\rangle.$

The affine variety of $I_{A}$ is

$\displaystyle\mathcal{V}(I_{A}):=\\{$ $\displaystyle
B=(b_{ij})\in\mathcal{M}_{n}\,:$ $\displaystyle
p_{A,2}(b_{11},\ldots,b_{nn})=\cdots=p_{A,n}(b_{11},\ldots,b_{nn})=0\\},$

and this is the set of matrices $B\in\mathcal{M}_{n}$ that are in additive FFP
with $A$:

$\mathcal{V}(I_{A})=\\{B\in\mathcal{M}_{n}\,:\,\chi_{A+B}(x)=\chi_{A}(x)\boxplus\chi_{B}(x)\\}.$

We denote by ${\\{A\\}}^{\boxplus}$ the variety $\mathcal{V}(I_{A})$. More
generally, we define the following.

######  Definition 4.1.

Given a set $\mathcal{A}\subseteq\mathcal{M}_{n}$, the additive finite free
complement or additive finite free variety ${\mathcal{A}}^{\boxplus}$ is the
set

${\mathcal{A}}^{\boxplus}:=\mathcal{V}(I_{\mathcal{A}})=\\{B\in\mathcal{M}_{n}\,:\,B\in\mathcal{V}(I_{A}),\mbox{
for all }A\in\mathcal{A}\\}=\bigcap_{A\in\mathcal{A}}\mathcal{V}(I_{A}),$

which is the affine variety of the ideal $I_{\mathcal{A}}$ generated by the
set of polynomials $p_{A,k}(B_{11},\ldots,B_{nn})$, for $A\in\mathcal{A}$ and
$k=2,\ldots,n$.

######  Definition 4.2.

When ${\mathcal{V}}^{\boxplus}=\mathcal{W}$ and
${\mathcal{W}}^{\boxplus}=\mathcal{V}$, we say that the varieties
$\mathcal{V},\mathcal{W}$ form an additive finite free complementary pair (an
additive complementary pair, for short).

Note that when $\mathcal{W}={\mathcal{V}}^{\boxplus}$ then
$\mathcal{W}={\mathcal{W}}^{\boxplus\,\boxplus}$.

We show next that every additive finite free variety contains the scalar
matrices.

######  Proposition 4.3.

If $A$ and $B$ are in additive FFP then so are $A+\lambda I$ and $B$, for
every $\lambda\in\mathbb{C}$.

###### Proof.

Let $A,B\in\mathcal{M}_{n}$. Then, $\chi_{A+\lambda I}(x)=\det(xI-(A+\lambda
I)=\det((x-\lambda)I-A)=\chi_{A}(x-\lambda)$. Similarly, $\chi_{A+\lambda
I+B}(x)=\chi_{A+B}(x-\lambda)$. By assumption
$\chi_{A}(x)\boxplus\chi_{B}(x)=\chi_{A+B}(x)$ and it follows that

$\displaystyle\chi_{A+\lambda I}(x)$
$\displaystyle\boxplus\chi_{B}(x)=\frac{1}{n!}\sum_{k=0}^{n}\chi_{A+\lambda
I}^{(k)}(x)\chi_{B}^{(n-k)}(0)$
$\displaystyle=\frac{1}{n!}\sum_{k=0}^{n}\chi_{A}^{(k)}(x-\lambda)\chi_{B}^{(n-k)}(0)$
$\displaystyle=\chi_{A+B}(x-\lambda)=\chi_{A+\lambda I+B}(x).$

∎

######  Remark 4.4.

Proposition 3.2 raises the question whether other algebraic operations
preserve additive FFP also in higher dimensions. The following example shows
that the obvious candidates fail: The matrices

$A=\begin{bmatrix}1&0&0\\\ 0&2&0\\\ 0&0&3\end{bmatrix}\qquad
B=\begin{bmatrix}1&-1&0\\\ -1&13&-3\\\ 0&-3&1\end{bmatrix}$

are in additive FFP but the following pairs are not:

  1. (i)

$(\lambda A,B)$, unless $\lambda\in\\{0,1\\}$;

  2. (ii)

$(A^{2},B)$;

  3. (iii)

$(A^{-1},B)$.

######  Definition 4.5.

Let $\mathcal{A}\subseteq\mathcal{M}_{n}$ and let
$\mathcal{V}={\mathcal{A}}^{\boxplus\,\boxplus}$. Then we say that
$\mathcal{A}$ is a generating set of $\mathcal{V}$, written
$\mathcal{V}=\langle\mathcal{A}\rangle$. We define the rank of $\mathcal{V}$
to be the minimal cardinality of a generating set of $\mathcal{V}$.

######  Proposition 4.6.

Every additive finite free variety $\mathcal{V}\subseteq\mathcal{M}_{n}$ is of
finite rank. An upper bound for the rank is
$\displaystyle\sum_{k=1}^{n-1}k!{n\choose k}^{2}$.

###### Proof.

Let $\mathcal{A}$ be a generating set of $\mathcal{V}$, that is,
$\mathcal{V}={\mathcal{A}}^{\boxplus\,\boxplus}$, and let $I_{\mathcal{A}}$ be
the corresponding ideal. That is, $I_{\mathcal{A}}$ is the sum of the ideals
$I_{A}$, for $A\in\mathcal{A}$, where $I_{\mathcal{A}}$ is generated by the
finitely many polynomials $p_{A,k}(B_{11},\ldots,B_{nn})$. Since
$I_{\mathcal{A}}$ is an ideal in a Noetherian ring, it is finitely generated.
It follows that there exists a finite set
$\mathcal{A^{\prime}}\subseteq\mathcal{A}$, such that
$I_{\mathcal{A}}=I_{\mathcal{A^{\prime}}}$. But then
$\mathcal{V}={\mathcal{A}}^{\boxplus\,\boxplus}={\mathcal{A^{\prime}}}^{\boxplus\,\boxplus}$,
that is $\mathcal{V}=\langle\mathcal{A^{\prime}}\rangle$.

Each polynomial $p_{A,k}(B_{11},\ldots,B_{nn})$, for $k=2,\ldots,n$, is of
degree at most $k-1$, since the part that is of degree $k$ does not involve
the elements of $A$ and is the same in $c_{k}$ and $d_{k}$, and therefore
vanishes in $p_{A,k}$. Similarly, the part of degree 0 comes just from $A$ and
also vanishes in $p_{A,k}$. The monomials in $p_{A,k}(B_{11},\ldots,B_{nn})$
are then of degree $1\leq d\leq n-1$ and of the form
$B_{i_{1}j_{1}}B_{i_{2}j_{2}}\cdots B_{i_{l}j_{l}}$, where
$\\{i_{1},\ldots,i_{l}\\}$ and $\\{j_{1},\ldots,j_{l}\\}$ are subsets of size
$l$ of $\\{1,\ldots,n\\}$ (by the way the determinant is defined). It follows
that the generators of the ideal $I_{\mathcal{A}}$ span linearly a subspace of
a vector space of dimension

$\sum_{k=1}^{n-1}{n\choose k}\frac{n!}{(n-k)!}=\sum_{k=1}^{n-1}k!{n\choose
k}^{2},$

which gives an upper bound on the number of generators of $I_{\mathcal{A}}$
and, thus, on the rank of $\mathcal{A}$. ∎

######  Remark 4.7.

The examples given in this paper suggest that the varieties are probably of
rank much smaller than the bound given in Proposition 4.6, more likely of
order $\mathcal{O}(n^{2})$, e.g., the Krull dimension of the coordinate ring
$\mathbb{C}[B_{11},\ldots,B_{nn}]/I_{A}$, possibly even satisfying
$\mathrm{rank}(\mathcal{V})+\mathrm{rank}(\mathcal{W})=n^{2}-1$, for varieties
$\mathcal{V}$ and $\mathcal{W}$ that are complementary.

The set of additive finite free varieties forms a lattice
$(\mathbb{L}_{n},\vee,\wedge,\mathcal{I}_{n},\mathcal{M}_{n})$ under set
inclusion, where:

  * •

The set $\bot=\mathcal{S}_{n}$ of scalar matrices is the bottom element of
$\mathbb{L}$ and $\top=\mathcal{M}_{n}$ is the top element.

  * •

For every $\mathcal{V},\mathcal{W}$ in $\mathbb{L}_{n}$, their join
$\mathcal{V}\vee\mathcal{W}:=\langle\mathcal{V}\cup\mathcal{W}\rangle$ and
their meet $\mathcal{V}\wedge\mathcal{W}:=\mathcal{V}\cap\mathcal{W}$ are in
$\mathbb{L}_{n}$.

Some of the properties of the lattice $\mathbb{L}_{n}$ that are easy to verify
are:

  1. (1)

$\mathbb{L}_{n}$ is associative, commutative and distributive with respect to
$\vee$ and $\wedge$.

  2. (2)

$\mathcal{V}\wedge\bot=\bot$ and $\mathcal{V}\vee\bot=\mathcal{V}$, for every
$\mathcal{V}\in\mathbb{L}_{n}$.

  3. (3)

$\mathcal{V}\wedge\top=\mathcal{V}$ and $\mathcal{V}\vee\top=\top$, for every
$\mathcal{V}\in\mathbb{L}_{n}$.

  4. (4)

${(\mathcal{V}\vee\mathcal{W})}^{\boxplus}={\mathcal{V}}^{\boxplus}\wedge{\mathcal{W}}^{\boxplus}$
and
${(\mathcal{V}\wedge\mathcal{W})}^{\boxplus}={\mathcal{V}}^{\boxplus}\vee{\mathcal{W}}^{\boxplus}$,
for every $\mathcal{V},\mathcal{W}\in\mathbb{L}_{n}$ (De Morgan laws).

######  Remarks 4.8.

  1. (1)

The complement of a variety in $\mathbb{L}_{n}$ does not obey the standard
definition of a complement in a lattice, that is, it does not hold in general
that $\mathcal{V}\vee{\mathcal{V}}^{\boxplus}=\top$ and
$\mathcal{V}\wedge{\mathcal{V}}^{\boxplus}=\bot$.

  2. (2)

Since every additive finite free variety contains the scalar matrices, we
could have defined the lattice in the quotient space
$\mathbb{M}_{n}:=\mathcal{M}_{n}/\mathcal{S}_{n}$.

Let us look at the special case of $2\times 2$ matrices.

######  Proposition 4.9.

Let the varieties $\mathcal{V},\mathcal{W}\subseteq\mathbb{C}^{{2}\times{2}}$
form an additive complementary pair. Then

  1. (1)

$\mathcal{V},\mathcal{W}$ are subspaces of $\mathbb{C}^{{2}\times{2}}$;

  2. (2)

$\mathrm{rank}(\mathcal{V})+\mathrm{rank}(\mathcal{W})=3$.

###### Proof.

  1. (1)

By (3.1), the elements of the matrices $A\in\mathcal{V}$, that are in additive
FFP with a matrix $B=(b_{ij})\in\mathcal{W}$, are the solution set of the
linear homogeneous equation

(4.1)
$(b_{22}-b_{11})A_{11}-(b_{22}-b_{11})A_{22}-2_{21}A_{12}-2b_{12}A_{21}=0$

and thus form a subspace of $\mathbb{C}^{{2}\times{2}}$. It follows that
$\mathcal{V}$, and similarly $\mathcal{W}$, are subspaces of
$\mathbb{C}^{{2}\times{2}}$.

  2. (2)

For each matrix $B\in\mathcal{W}$, we obtain a linear equation of the form
(4.1). Hence, it is sufficient to find the matrices $A$ that are in additive
FFP with respect to a linear basis of the subspace $\mathcal{W}$. One of the
elements of $\mathcal{W}$ (written as a row vector) is
$(b_{11},b_{22},b_{12},b_{21})=(1,1,0,0)$, representing the identity matrix.
Thus, by reduction through this row, we may assume that the other basis
elements have zero in their first entry. When
$(b_{11},b_{22},b_{12},b_{21})=(1,1,0,0)$ then Equation (4.1) vanishes, and
so, we get that $\mathcal{V}$ is the set of solutions of an homogeneous system
of linear equations of the form

(4.2) $c_{1}A_{11}-c_{1}A_{22}+c_{2}A_{12}+c_{3}A_{21}=0.$

The number of these equations is $\dim(\mathcal{W})-1$ (since the identity
matrix does not contribute an equation). Moreover, since these equations are
independent, we cannot take a smaller number of equations in order to define
$\mathcal{V}$. A basis for $\mathcal{V}$ (a basis for the solution of the
system of equations) consists of $4-(\dim(\mathcal{W})-1)=5-\dim(\mathcal{W})$
elements. It follows from the above discussion that
$\mathrm{rank}(\mathcal{V})=\dim(\mathcal{V})-1$,
$\mathrm{rank}(\mathcal{W})=\dim(\mathcal{W})-1$ (after omitting the identity
matrix), and

$\mathrm{rank}(\mathcal{V})+\mathrm{rank}(\mathcal{W})=5-2=3.$

∎

##  5\. Additive finite free complementary pairs

In this section we present pairs of varieties
$(\mathcal{V},\mathcal{W})\subseteq\mathcal{M}_{n}\times\mathcal{M}_{n}$ that
form additive complementary pairs. These are:
$(\mathcal{D}_{n},\mathcal{B}_{n})$, the set of diagonal matrices and the set
of principally balanced matrices;
$(\widehat{\mathcal{R}}_{n},\mathcal{R}_{n})$ (resp.
$(\widehat{\mathcal{L}}_{n},\mathcal{L}_{n})$), the set of upper (resp. lower)
triangular matrices with constant diagonal and the set of upper (resp. lower)
triangular matrices; $(\mathcal{S}_{n},\mathcal{M}_{n})$, the set of scalar
matrices and the set of all $n\times n$ matrices.

We will make frequent use of the following lemma. We denote by
$E_{kl}\in\mathcal{M}_{n}$ the matrix with 1 at position $(k,l)$ and 0
elsewhere.

######  Lemma 5.1.

Let $A=(a_{ij})\in\mathcal{M}_{n}$ be a matrix having an off-diagonal non-zero
entry $a_{lk}$. Then
$\chi_{A+E_{kl}}(x)\neq\chi_{A}(x)=\chi_{A}(x)\boxplus\chi_{E_{kl}}(x)$.

###### Proof.

Let us look at the coefficient of $x^{n-2}$ in $\chi_{A+E_{kl}}(x)$ and in
$\chi_{A}(x)$. The only term that is affected by the addition of $E_{kl}$ to
$A$ is the principal minor with indices $k,l$. The difference is

$(a_{kk}a_{ll}-(a_{kl}+1)a_{lk})-(a_{kk}a_{ll}-a_{kl}a_{lk})=-a_{lk}\neq 0.$

It follows that $\chi_{A+E_{kl}}(x)\neq\chi_{A}(x)$. On the other hand,
$\chi_{E_{kl}}(x)=x^{n}$ and by Example 2.2,
$\chi_{A}(x)\boxplus\chi_{E_{kl}}(x)=\chi_{A}(x)$. ∎

###  5.1. Diagonal, principally balanced and symmetric matrices

Consider first diagonalizable matrices. Because of invariance under
conjugation (Remark 4 (4)) it we may assume that one of the matrices is
diagonal. It turns out that the following concept is crucial.

######  Definition 5.2.

A matrix $B\in\mathcal{M}_{n}$ is called principally balanced (or has the
symmetrized principal minors property [6]) if, for every $i$, $1\leq i\leq n$,
the values of all principal minors of $B$ of order $i$ coincide. We denote by
$\mathcal{B}_{n}\subseteq\mathcal{M}_{n}$ the family of $n\times n$ matrices
which are principally balanced.

This is a special case of the principal minor assignment problem [4, 11] going
back to Stouffer [15].

It it easy to see that the family of principally balanced matrices is
invariant under conjugation by permutation matrices and diagonal matrices.
Clearly, all $1\times 1$ matrices are principally balanced, as well as all
$2\times 2$ matrices with a constant diagonal and all $n\times n$ triangular
matrices with a constant diagonal. Here are examples of specific principally
balanced matrices.

######  Example 5.3.

The following $3\times 3$ matrix is principally balanced: all principal minors
of order $1$ (the diagonal entries) equal $1$ and all principal minors of
order $2$ equal $-11$:

$\begin{bmatrix}[r]1&2&3\\\ 6&1&-12\\\ 4&-1&1\end{bmatrix}.$

######  Example 5.4.

Let $A$ be the the $n\times n$ matrix with entries $a_{ij}=\frac{i}{j}$,
$i=1,\ldots,n$:

$A=\begin{bmatrix}[r]1&\frac{1}{2}&\frac{1}{3}\cdots&\frac{1}{n}\\\\[2.84526pt]
2&1&\frac{2}{3}\cdots&\frac{2}{n}\\\\[2.84526pt]
3&\frac{3}{2}&1\cdots&\frac{3}{n}\\\\[-2.84526pt]
\vdots&&\ddots&\vdots\\\\[-2.84526pt]
n&\frac{n}{2}&\frac{n}{3}\cdots&1\end{bmatrix}.$

The rank of $A$ is 1, so all minors (not just principal) of order $k\geq 2$
equal 0, while the principal minors of order 1 equal 1. Thus, $A$ is
principally balanced.

More generally, let $A_{i}$ denote the $i$-th row of $A$, $i=1,\ldots,n$. When
all entries of $A_{1}$ are non-zero and $A_{i}=\frac{a_{11}}{a_{1i}}A_{1}$,
$i=1,\ldots,n$, then $A$ is principally balanced with all principal minors of
order $1$ equal $a_{11}$ and all minors of order $k\geq 2$ equal $0$. In fact,
it is easy to show that for any positive integer $k$, $A^{k}=(a_{11}n)^{k-1}A$
and so, for any polynomial $p(x)$ with $p(0)=0$, $p(A)$ is a scalar multiple
of $A$ and thus of the same form as $A$ (just that the principal minors of
order 1 are multiplied by $\lambda$) and, in particular, is principally
balanced. Moreover, $p(A)$ is principally balanced for every polynomial $P$
since the set of principally balanced matrices is closed under addition of a
scalar matrix by Proposition 4.3 and Theorem 5.7.

######  Lemma 5.5.

Let $B\in\mathcal{B}_{n}$ be principally balanced and let $m_{i}$, for
$i=0,\ldots,n$, be its $i\times i$ principal minor, where $m_{0}$ is defined
to be $1$. Then $\chi_{B}(x)=\sum_{i=0}^{n}(-1)^{i}{n\choose i}m_{i}x^{n-i}$.

###### Proof.

For each $i$, the coefficient of $x^{n-i}$ in $\chi_{B}(x)$ is $(-1)^{i}$
times the sum of all principal minors of order $i$. There are ${n\choose i}$
such principal minors and each one equals $m_{i}$. ∎

######  Remark 5.6.

The family of principally balanced matrices can be equivalently defined by the
property of having equal “cycle sums” of the same order: $c_{I}=c_{J}$, for
all $I,J\subseteq[n]$ with $|I|=|J|$, where

$c_{I}=\sum_{\begin{subarray}{c}I=\\{i_{1},\ldots,i_{k}\\}\\\ i_{1}=\min
I\end{subarray}}a_{i_{1},i_{2}}\cdots a_{i_{k-1},i_{k}}a_{i_{k},i_{1}},$

are the cycle sums of a matrix $A=(a_{ij})$, see [6]. Indeed the authors prove
that principal minors and cycle sums are related by Möbius inversion on the
lattice of set partitions.

This can be interpreted as follows. Let $B\in\mathcal{B}_{n}$ be a principally
balanced matrix with principal minors $m_{k}$ and cycle sums $c_{k}$. Consider
$\tilde{m}_{k}=(-1)^{k}m_{k}$ as moments of a formal random variable (or
“umbra” in the sense of Rota [13]) $Z$, then

$\chi_{B}(x)=\mathbb{E}(x+Z)^{n}$

and the corresponding classical cumulants of $Z$ are the numbers
$\tilde{c}_{k}=(-1)^{k}c_{k}$.

We denote by $[n]$ the set $\\{1,\ldots,n\\}$. When $M\in\mathcal{M}_{n}$ and
$S\subseteq[n]$ with $|S|=k$, we denote by $M_{S}$ the $k\times k$ principal
submatrix of $M$ obtained after removing the rows and columns with indices in
$[n]-S$.

######  Theorem 5.7.

Let $\mathcal{D}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
diagonal matrices and let $\mathcal{B}_{n}\subseteq\mathcal{M}_{n}$ be the
family of $n\times n$ principally balanced matrices. Then $\mathcal{D}_{n}$
and $\mathcal{B}_{n}$ form an additive complementary pair.

###### Proof.

The theorem clearly holds for $n=1$, so let us assume that $n\geq 2$. First,
we show that for arbitrary $D\in\mathcal{D}_{n}$ and $B\in\mathcal{B}_{n}$,
the equality $\chi_{D+B}(x)=\chi_{D}(x)\boxplus\chi_{B}(x)$ holds. Let
$D=\mathrm{diag}(d_{1},\ldots,d_{n})$, hence

$\displaystyle\chi_{D}(x)$
$\displaystyle=\sum_{i=0}^{n}c_{i}x^{n-i}=\prod_{i=1}^{n}(x-d_{i})$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}\det(D_{J})\bigg{)}x^{n-i}$
(5.1)
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}d_{J}\bigg{)}x^{n-i},$

where $d_{J}=d_{j_{1}}\cdots d_{j_{i}}$ for $J=\\{j_{1},\ldots,j_{i}\\}$.

For $B\in\mathcal{B}_{n}$, we have

(5.2) $\chi_{B}(x)=\sum_{i=0}^{n}b_{i}x^{n-i}=\sum_{i=0}^{n}(-1)^{i}{n\choose
i}m_{i}x^{n-i},$

where $m_{i}$, $i=0,\ldots,n$, is the value of a principal minor of $B$ of
order $i$.

The characteristic polynomial of $B+D$ is

$\displaystyle\chi_{D+B}(x)$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{I\subseteq[n],\,|I|=i}\det((D+B)_{I})\bigg{)}x^{n-i}$
(5.3)
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{I\subseteq[n],\,|I|=i}\sum_{j=0}^{i}\sum_{J\subseteq
I,\,|J|=j}m_{i-j}d_{J}\bigg{)}x^{n-i}.$

We observe that when we compute $\det((D+B)_{I})$, with $|I|=i$, in (5.1),
then we sum over the products of $d_{J}$, which is a principal minor of order
$j\leq i$ of $D$, and a principal minor of order $i-j$ of $B$. But all
principal minors of order $i-j$ of $B$ equal $m_{i-j}$. It follows that for
each $J$, the term $m_{i-j}d_{J}$ appears $n-j\choose i-j$ times in (5.1), the
number of principal minors of order $i-j$ of $B$ with indices of the
submatrices outside of $J$, while the submatrix $D_{J}$ of $D$ is kept fixed.
Hence

$\displaystyle\chi_{D+B}(x)$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{j=0}^{i}\sum_{J\subseteq[n],\,|J|=j}{n-j\choose
i-j}m_{i-j}d_{J}\bigg{)}x^{n-i}$ (5.4)
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{j=0}^{i}{n-j\choose
i-j}m_{i-j}\sum_{J\subseteq[n],\,|J|=j}d_{J}\bigg{)}x^{n-i}.$

Next, we compute the additive convolution of $\chi_{D}(x)$ and $\chi_{B}(x)$.
By (2.1),

(5.5) $\displaystyle\chi_{D}(x)\boxplus\chi_{B}(x)$
$\displaystyle=\sum_{i=0}^{n}f_{i}x^{n-i}=\sum_{i=0}^{n}c_{i}x^{n-i}\boxplus\sum_{i=0}^{n}b_{i}x^{n-i}$
(5.6)
$\displaystyle=\sum_{i=0}^{n}\bigg{(}\sum_{j=0}^{i}\frac{(n-j)!(n-i+j)!}{n!(n-i)!}c_{j}b_{i-j}\bigg{)}x^{n-i}.$

Substituting for $c_{j}$ the appropriate coefficient from (5.1) and for
$b_{i-j}$ the appropriate coefficient from (5.2), we get

$\displaystyle f_{i}$
$\displaystyle=\sum_{j=0}^{i}\frac{(n-j)!(n-i+j)!}{n!(n-i)!}(-1)^{j}\sum_{J\subseteq[n],\,|J|=j}d_{J}(-1)^{i-j}{n\choose
i-j}m_{i-j}$
$\displaystyle=(-1)^{i}\sum_{j=0}^{i}\frac{(n-j)!(n-i+j)!n!}{n!(n-i)!(n-i+j)!(i-j)!}m_{i-j}\sum_{J\subseteq[n],\,|J|=j}d_{J}$
(5.7) $\displaystyle=(-1)^{i}\sum_{j=0}^{i}{n-j\choose
i-j}m_{i-j}\sum_{J\subseteq[n],\,|J|=j}d_{J}.$

We got in (5.1) the same coefficient of $x^{n-i}$ as in (5.1), hence
$\chi_{D+B}(x)=\chi_{D}(x)\boxplus\chi_{B}(x)$. It follows that
$\mathcal{B}_{n}\subseteq{\mathcal{D}_{n}}^{\boxplus}$ and
$\mathcal{D}_{n}\subseteq{\mathcal{B}_{n}}^{\boxplus}$.

We show now that ${\mathcal{D}_{n}}^{\boxplus}\subseteq\mathcal{B}_{n}$. Let
$B=(b_{ij})\in{\mathcal{D}_{n}}^{\boxplus}$. Let $K\subseteq[n]$, $|K|=k$,
$0<k<n$, and let $D(\lambda,K)=(d_{ij})$ be the parametric diagonal matrix
with entries

$d_{ij}=\left\\{\begin{array}[]{rl}\lambda&\mbox{for }i=j\in K,\\\
0&\mbox{otherwise.}\end{array}\right.$

Evaluating $\chi_{D(\lambda,K)+B}(x)$ at $x=0$ gives

$\chi_{D(\lambda,K)+B}(0)=(-1)^{n}\det(D(\lambda,K)+B)=(-1)^{n}\det(B_{[n]-K})\lambda^{k}+q(\lambda),$

where $q(\lambda)$ is a polynomial in $\lambda$ of degree less than $k$. It
follows that

(5.8)
$\lim_{\lambda\to\infty}\frac{\chi_{D(\lambda,K)+B}(0)}{\lambda^{k}}=(-1)^{n}\det(B_{[n]-K}).$

Let us now compute this limit for $\chi_{D(\lambda,K)}(x)\boxplus\chi_{B}(x)$.
The characteristic polynomial of $D(\lambda,K)$ is

$\chi_{D(\lambda,K)}(x)=\sum_{i=0}^{n}a_{i}x^{n-i}=x^{n-k}(x-\lambda)^{k}=\sum_{i=0}^{n}{k\choose
i}(-\lambda)^{i}x^{n-i},$

where ${k\choose i}=0$ when $i>k$. That is,

$a_{i}=\left\\{\begin{array}[]{ccl}(-1)^{i}{k\choose
i}\lambda^{i}&\mbox{for}&0\leq i\leq k,\\\
0&\mbox{for}&i>k.\end{array}\right.$

The characteristic polynomial of $B$ is

$\chi_{B}(x)=\sum_{i=0}^{n}b_{i}x^{n-i}=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{I\subseteq[n],\,|I|=i}\det(B_{I})\bigg{)}x^{n-i}.$

That is, $(-1)^{i}b_{i}$ is the sum of the ${n\choose i}$ principal minors of
order $i$.

By (2.1),

$\displaystyle(\chi_{D(\lambda,K)}\boxplus\chi_{B})(0)$
$\displaystyle=\sum_{i=0}^{n}\frac{(n-i)!i!}{n!}a_{i}b_{n-i}$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}\frac{(-1)^{i}{k\choose
i}(-1)^{n-i}}{{n\choose
i}}\sum_{I\subseteq[n],\,|I|=n-i}\det(B_{I})\bigg{)}\lambda^{i},$

a polynomial of degree at most $k$ in $\lambda$. It follows that

$(\chi_{D(\lambda,K)}\boxplus\chi_{B})(0)=\bigg{(}\frac{(-1)^{n}}{{n\choose
k}}\sum_{I\subseteq[n],\,|I|=n-k}\det(B_{I})\bigg{)}\lambda^{k}+p(\lambda),$

where $p(\lambda)$ is a polynomial in $\lambda$ of degree less than $k$. Then,

(5.9)
$\lim_{\lambda\to\infty}\frac{(\chi_{D(\lambda,K)}\boxplus\chi_{B})(0)}{\lambda^{k}}=\frac{(-1)^{n}}{{n\choose
k}}\sum_{I\subseteq[n],\,|I|=n-k}\det(B_{I}).$

Comparing (5.8) with (5.9), we get that

$\det(B_{[n]-K})=\frac{1}{{n\choose
k}}\sum_{I\subseteq[n],\,|I|=n-k}\det(B_{I}),$

that is, for every $1\leq k\leq n$, every principal minor of $B$ of order
$n-k$ equals the mean value of the principal minors of order $n-k$. In other
words, all principal minors of the same order are equal. It follows that $B$
is principally balanced and
${\mathcal{D}_{n}}^{\boxplus}\subseteq\mathcal{B}_{n}$. Together with the
inclusion $\mathcal{B}_{n}\subseteq{\mathcal{D}_{n}}^{\boxplus}$ we have

${\mathcal{D}_{n}}^{\boxplus}=\mathcal{B}_{n}.$

It remains to show that the additive free complement of $\mathcal{B}_{n}$ is
$\mathcal{D}_{n}$ and not a larger family. Suppose that $A$ has an off-
diagonal entry $a_{kl}\neq 0$, $k\neq l$ Then the matrix $E_{kl}$ from Lemma
5.1 satisfies that
$\chi_{A+E_{kl}}(x)\neq\chi_{A}(x)\boxplus\chi_{E_{kl}}(x)$. On the other hand
all principal minors of $E_{kl}$ vanish and hence $E_{kl}\in\mathcal{B}_{n}$.
We showed that $A\notin{\mathcal{B}_{n}}^{\boxplus}$ and consequently
${\mathcal{B}_{n}}^{\boxplus}\subseteq\mathcal{D}_{n}$. Together with the
inclusion in the other direction, we have

${\mathcal{B}_{n}}^{\boxplus}=\mathcal{D}_{n}$

and the proof is complete. ∎

######  Corollary 5.8.

Let $\mathcal{A}_{n}\subseteq\mathbb{R}^{n\times n}$ be a maximal family of
commuting diagonalizable matrices and let $\mathcal{B}_{n}$ be the family of
principally balanced matrices.

  1. (1)

There exists a non-singular matrix $P$, such that $\mathcal{A}_{n}$ and
$P^{-1}{\mathcal{B}_{n}}P$ form an additive complementary pair.

  2. (2)

If $\mathcal{A}_{n}$ is a maximal family of commuting symmetric matrices then
there exists an orthogonal matrix $U$, such that $\mathcal{A}_{n}$ and
$U^{T}{\mathcal{B}_{n}}U$ form an additive complementary pair.

###### Proof.

Since the matrices in $\mathcal{A}_{n}$ commute with each other, they are
simultaneously diagonalizable ([5], Theorem 1.3.21). That is, there exists a
non-singular matrix $P$, such that for each $A\in\mathcal{A}_{n}$ the
conjugated matrix $D=PAP^{-1}$ is diagonal. By the maximality of
$\mathcal{A}_{n}$ and the non-singularity of $P$,
$\mathcal{A}_{n}=P^{-1}\mathcal{D}_{n}P$ is a subspace of dimension $n$. By
Theorem 5.7, for every $A\in\mathcal{A}_{n}$ and $B\in\mathcal{B}_{n}$,

$\chi_{PAP^{-1}+B}(x)=\chi_{PAP^{-1}}(x)\boxplus\chi_{B}(x)$

and the subsets $P\mathcal{A}_{n}P^{-1}$ and $\mathcal{B}_{n}$ form an
additive complementary pair. By similarity transformation, the same holds for
$\mathcal{A}_{n}$ and $P^{-1}{\mathcal{B}_{n}}P$.

If the matrices in $\mathcal{A}_{n}$ are symmetric then commuting is
equivalent to simultaneous orthogonal diagonalization ([5, Theorem 4.5.15]). ∎

###  5.2. Scalar matrices

Another additive complementary pair is the following.

######  Theorem 5.9.

Let $\mathcal{S}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
scalar matrices. Then $\mathcal{S}_{n}$ and $\mathcal{M}_{n}$ form an additive
complementary pair.

###### Proof.

Let $A\in\mathcal{M}_{n}$ be an arbitrary matrix. Then the zero matrix is in
finite free position with $A$ because $\chi_{0}(x)=x^{n}$ is the neutral
element for additive finite free convolution (Example 2.2). It follows from
Proposition 4.3 that any scalar matrix is in finite free position with $A$ as
well. It follows that

$\mathcal{S}_{n}\subseteq{\mathcal{M}_{n}}^{\boxplus}.$

For the converse, by Theorem 5.7, since
$\mathcal{M}_{n}\supseteq\mathcal{D}_{n}$ and
$\mathcal{M}_{n}\supseteq\mathcal{B}_{n}$,

${\mathcal{M}_{n}}^{\boxplus}\subseteq{\mathcal{D}_{n}}^{\boxplus}\cap{\mathcal{B}_{n}}^{\boxplus}=\mathcal{B}_{n}\cap\mathcal{D}_{n}=\mathcal{S}_{n}.$

By both inclusions, we conclude that

${\mathcal{M}_{n}}^{\boxplus}=\mathcal{S}_{n}.$

Clearly, since
${\mathcal{S}_{n}}^{\boxplus}={\mathcal{M}_{n}}^{\boxplus\,\boxplus}\supseteq\mathcal{M}_{n}$,
we get that

${\mathcal{S}_{n}}^{\boxplus}=\mathcal{M}_{n}.$

It follows that $\mathcal{S}_{n}$ and $\mathcal{M}_{n}$ form an additive
complementary pair. ∎

###  5.3. Triangular matrices

In this subsection we exhibit an additive complementary pair within the ring
of upper triangular matrices.

######  Theorem 5.10.

Let $\mathcal{R}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
upper triangular matrices and let
$\widehat{\mathcal{R}}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times
n$ upper triangular matrices with constant diagonal. Then $\mathcal{R}_{n}$
and $\widehat{\mathcal{R}}_{n}$ form an additive complementary pair.

Analogously, the lower triangular matrices $\mathcal{L}_{n}$ and
$\widehat{\mathcal{L}}_{n}$ form an additive complementary pair.

###### Proof.

Let $T\in\mathcal{R}_{n}$ and $C\in\widehat{\mathcal{R}}_{n}$. Then
$\chi_{T}(x),\chi_{C}(x)$ and $\chi_{T+C}(x)$ do not change if we replace them
by their diagonals, i.e., if we set all off-diagonal entries to zero. Then $T$
becomes diagonal and $C$ becomes a scalar matrix and thus principally
balanced. By Theorem 5.7, $\chi_{T+C}(x)=\chi_{T}(x)\boxplus\chi_{C}(x)$. This
shows that $\widehat{\mathcal{R}}_{n}\subseteq{\mathcal{R}_{n}}^{\boxplus}$
and $\mathcal{R}_{n}\subseteq{\widehat{\mathcal{R}}_{n}}^{\boxplus}$.

Next, we show that
${\widehat{\mathcal{R}}_{n}}^{\boxplus}\subseteq\mathcal{R}_{n}$. Suppose that
$A=(a_{ij})\notin\mathcal{R}_{n}$, i.e., $a_{lk}\neq 0$ for some $(l,k)$ with
$l>k$. Then the matrix $E_{kl}$ from Lemma 5.1 is in
$\widehat{\mathcal{R}}_{n}$ but
$\chi_{A+E_{kl}}(x)\neq\chi_{A}(x)\boxplus\chi_{E_{kl}}(x)$. It follows that
$A\notin{\widehat{\mathcal{R}}_{n}}^{\boxplus}$ and consequently that
${\widehat{\mathcal{R}}_{n}}^{\boxplus}\subseteq\mathcal{R}_{n}$.

It remains to show that
${\mathcal{R}_{n}}^{\boxplus}\subseteq\widehat{\mathcal{R}}_{n}$. Let
$C\in{\mathcal{R}_{n}}^{\boxplus}$, then the argument of the previous
paragraph shows that $C$ is upper triangular. Since
$\mathcal{D}_{n}\subseteq\mathcal{R}_{n}$, it follows from Theorem 5.7 that
$C$ is principally balanced and thus
$C\in\mathcal{R}_{n}\cap\mathcal{B}_{n}=\widehat{\mathcal{R}}_{n}$.

∎

######  Corollary 5.11.

If $A\in\mathcal{M}_{n}$ has a single eigenvalue and $B\in\mathcal{M}_{n}$
commutes with $A$ then, for any polynomials $p(x)$ and $q(x)$, $p(A)$ and
$q(B)$ are in additive FFP.

###### Proof.

Since $A$ and $B$ commute, they are simultaneously triangularizable [5,
Theorem 2.3.3], that is, for some regular matrix $P$, $P^{T}AP$ and $P^{T}BP$
are upper triangular, with $P^{T}AP$ having a constant diagonal. The same
holds for $p(A)$ and $q(B)$, so, by Theorem 5.10, $p(A)$ and $q(B)$ are in
additive FFP. ∎

######  Remark 5.12.

Unlike diagonalization, commuting matrices are simultaneously
triangularizable, but the converse is not true in general.

###  5.4. Conclusion

We conclude this section with the following observations. Denote
$\widehat{\mathcal{V}}=\mathcal{V}\cap\mathcal{B}_{n}$ for any subset
$\mathcal{V}\subseteq\mathcal{M}_{n}$. In particular,
$\mathcal{B}_{n}=\widehat{\mathcal{M}}_{n}$.

  1. (1)

This notation is compatible with Theorem 5.10 by the observation that a
triangular matrix is principally balanced if and only if it has constant
diagonal.

  2. (2)

In particular, scalar matrices are exactly the principally balanced diagonal
matrices, $\widehat{\mathcal{D}}_{n}=\mathcal{S}_{n}$.

  3. (3)

The examples found so far can be summarized in the following “recipe”:

    1. 1.

Pick a pair $(\mathcal{V}_{1},\mathcal{V}_{2})$ among
$(\mathcal{D}_{n},\mathcal{M}_{n})$, $(\mathcal{R}_{n},\mathcal{R}_{n})$ or
$(\mathcal{L}_{n},\mathcal{L}_{n})$.

    2. 2.

Restrict one of the components to its principally balanced subset
$\widehat{\mathcal{V}}_{i}$.

##  6\. Moments of sums of matrices that are in additive FFP

In [2] Arizmendi and Perales introduced cumulants for the additive finite free
convolution as the coefficients of a truncated $R$-transform, by showing that
they satisfy the axiomatization of cumulants as defined by Lehner in [7].
Asymptotically, the finite free cumulants converge to free cumulants. Finite
free cumulants are additive with respect to finite free convolution [2], which
implies that for matrices $A$ and $B$ in additive FFP we have

(6.1) $\kappa_{i}(A+B)=\kappa_{i}(A)+\kappa_{i}(B),$

where $\kappa_{i}(A)$ is the $i$-th order additive finite free cumulant of
(the characteristic polynomial of) $A$. A similar equality holds for
independent as well as for free independent random variables. However, unlike
what happens in probability and in free probability, where mixed cumulants of
independent (free) random variables vanish, this is not the case for finite
free cumulants.

Cumulants are tightly related to moments, with concrete moment-cumulant
formulas, which allows passing from one representation to the other. Such
formulas were introduced in [2] in the finite free setting. This means that
one can obtain formulas for the moments of $A+B$ in terms of the moments of
$A$ and those of $B$ when $A$ and $B$ are in finite free position, as we show
below. Conversely, when $A$ and $B$ satisfy these moment formulas, for
$k=2,\ldots,n$, then they are in additive free position.

The first moment (or normalized trace) of $A\in\mathcal{M}_{n}$ is defined to
be

$m_{1}(A):=\operatorname{\mathrm{tr}}(A)=\frac{1}{n}\operatorname{\mathrm{Tr}}(A)=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i},$

the arithmetic mean of the eigenvalues $\lambda_{i}$ of $A$. In general, the
$k$-th moment of $A$, $k\geq 1$, is

$m_{k}(A):=\operatorname{\mathrm{tr}}(A^{k})=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i}^{k}.$

We make use of the following coefficient-moment formula of Lewin [8] that is
based on Newton’s identities. Let $\chi_{A}(x)=\sum_{k=0}^{n}a_{k}x^{n-k}$.
Then, $a_{0}=1$, and for $k=1,\ldots,n$,

(6.2) $a_{k}=\sum_{\pi}\prod_{i=1}^{t}\frac{(-n\cdot
m_{r_{i}}(A))^{s_{i}}}{r_{i}^{s_{i}}s_{i}!},$

where the summation is over all partitions $\pi$ of $k$ of the form

$k=\underbrace{r_{1}+\cdots+r_{1}}_{s_{1}\,\mathrm{summands}}+\underbrace{r_{2}+\cdots+r_{2}}_{s_{2}\,\mathrm{summands}}+\cdots+\underbrace{r_{t}+\cdots+r_{t}}_{s_{t}\,\mathrm{summands}}=\sum_{i=1}^{t}s_{i}r_{i},$

with $0<r_{1}<r_{2}<\cdots<r_{t}$, and
$r_{i}=r_{i}(\pi),s_{i}=s_{i}(\pi),t=t(\pi)$.

######  Theorem 6.1.

Let $A,B\in\mathcal{M}_{n}$ be in additive FFP. Then the moments of $A+B$ can
be expressed in terms of the moments of $A$ and the moments of $B$.

###### Proof.

The proof is by induction. For $r=1$, we have

$m_{1}(A+B)=m_{1}(A)+m_{1}(B)$

by linearity of the trace operator. Suppose that the statement holds for
$1\leq i<r$. Let
$\chi_{A+B}(x)=\chi_{A}(x)\boxplus\chi_{B}(x)=\sum_{i=0}^{n}c_{i}x^{n-i}$. By
Newton’s identity,

$-rc_{r}=\sum_{i=0}^{r-1}c_{i}\operatorname{\mathrm{Tr}}(A+B)^{r-i},$

thus,

(6.3) $m_{r}(A+B)=-\frac{r}{n}c_{r}-\sum_{i=1}^{r-1}c_{i}m_{r-i}(A+B).$

Since $A$ and $B$ are in additive FFP, we have, for $1\leq i\leq r$,

$c_{i}=\sum_{l+j=i}\frac{{n-l\choose j}}{{n\choose j}}a_{l}b_{j},$

and by (6.2), the coefficients $a_{l}$ and $b_{j}$ can be written in terms of
the moments of $A$, respectively $B$. It remains to express the moments
$m_{r-i}(A+B)$ on the right hand side of (6.3) through the moments of $A$ and
$B$, which is assumed by the induction hypothesis.

Equivalently, we can obtain a complex formula for $m_{r}(A+B)$ through the
formulas in [2]. By Equation (4.5) in [2], $m_{r}(A+B)$ can be written in
terms of the cumulants $\kappa_{1}(A+B),\ldots,\kappa_{r}(A+B)$. Then, as
shown in [2], Proposition 3.6, finite free cumulants are additive with respect
to polynomial convolution, so that for each $i$,
$\kappa_{i}(A+B)=\kappa_{i}(A)+\kappa_{i}(B)$. Finally, by Equation (4.4) in
[2], each $\kappa_{i}(A)$ can be written as an expression in the moments
$m_{1}(A),\ldots,m_{i}(A)$, and similarly for $\kappa_{i}(B)$. ∎

######  Examples 6.2.

We obtain simple formulas for the first, second and third moments of $A+B$
when $A,B\in\mathcal{M}_{n}$ are in additive FFP. For the higher moments the
formulas become more complex and, unlike the case $n\leq 3$, they depend on
the dimension of the matrices. Here are the first four moments.

$\displaystyle m_{1}(A+B)$ $\displaystyle=m_{1}(A)+m_{1}(B);$ $\displaystyle
m_{2}(A+B)$ $\displaystyle=m_{2}(A)+2m_{1}(A)m_{1}(B)+m_{2}(B);$
$\displaystyle m_{3}(A+B)$
$\displaystyle=m_{3}(A)+3m_{2}(A)m_{1}(B)+3m_{1}(A)m_{2}(B)+m_{3}(B);$
$\displaystyle m_{4}(A+B)$
$\displaystyle=m_{4}(A)+4m_{3}(A)m_{1}(B)+\mbox{\large$\frac{2n}{n-1}$}m_{2}(A)m_{1}^{2}(B)$
$\displaystyle\quad+\mbox{\large$\frac{4n-6}{n-1}$}m_{2}(A)m_{2}(B)-\mbox{\large$\frac{2n}{n-1}$}m_{1}^{2}(A)m_{1}^{2}(B)$
$\displaystyle\quad+\mbox{\large$\frac{2n}{n-1}$}m_{1}^{2}(A)m_{2}(B)+4m_{1}(A)m_{3}(B)+m_{4}(B).$

We demonstrate the above formula for the second moment.

$\displaystyle m_{2}(A+B)=-\mbox{\large$\frac{2}{n}$}c_{2}-c_{1}m_{1}(A+B)$
$\displaystyle=-\mbox{\large$\frac{2}{n}$}\Big{(}a_{0}b_{2}+\mbox{\large$\frac{n-1}{n}$}a_{1}b_{1}+a_{2}b_{0}\Big{)}-\Big{(}(a_{0}b_{1}+a_{1}b_{0})m_{1}(A+B)\Big{)}$
$\displaystyle=-\mbox{\large$\frac{2}{n}$}\Big{(}(\mbox{\large$\frac{n^{2}}{2}$}m_{1}^{2}(B)-\mbox{\large$\frac{n}{2}$}m_{2}(B))+\mbox{\large$\frac{n-1}{n}$}n^{2}m_{1}(A)m_{1}(B)\Big{.}$
$\displaystyle\quad+\Big{.}(\mbox{\large$\frac{n^{2}}{2}$}m_{1}^{2}(A)-\mbox{\large$\frac{n}{2}$}m_{2}(A))\Big{)}$
$\displaystyle\quad+\Big{(}n(m_{1}(A)+m_{1}(B))(m_{1}(A)+m_{1}(B))\Big{)}$
$\displaystyle=-nm_{1}^{2}(B)+m_{2}(B)-2(n-1)m_{1}(A)m_{1}(B)-nm_{1}^{2}(A)+m_{2}(A)$
$\displaystyle\quad+nm_{1}^{2}(B)+2nm_{1}(A)m_{1}(B)+nm_{1}^{2}(A)$
$\displaystyle=m_{2}(A)+2m_{1}(A)m_{1}(B)+m_{2}(B).$

By the linearity of the moment and the fact that
$\operatorname{\mathrm{Tr}}(AB)=\operatorname{\mathrm{Tr}}(BA)$, it follows
from the formulas for the second and third moments that when $A$ and $B$ are
in FFP then:

$\displaystyle m_{1}(AB)=m_{1}(A)m_{1}(B);$ $\displaystyle
m_{1}(AB^{2})+m_{1}(A^{2}B)=m_{1}(A)m_{2}(B)+m_{2}(A)m_{1}(B).$

Let us now examine the case when $A$ is in additive FFP with itself. We denote
by $\mathcal{P}(j)$ the set of partitions of the set $[j]=\\{1,\ldots,j\\}$.
$\mathcal{P}(j)$ forms a lattice with a least element
$0_{j}=\\{\\{1\\},\\{2\\},\ldots,\\{j\\}\\}$ and an upper element
$1_{j}=\\{\\{1,2,\ldots,j\\}\\}$. When
$\pi=\\{V_{1},\ldots,V_{r}\\}\in\mathcal{P}(j)$ is a partition of $[j]$ with
$r$ blocks then we set $|\pi|=r$ and denote by $\kappa_{\pi}(A)$ the product
of the cumulants

$\kappa_{\pi}(A):=\kappa_{|V_{1}|}(A)\kappa_{|V_{2}|}(A)\cdots\kappa_{|V_{r}|}(A).$

The Möbius function $\mu(0_{j},\pi)$ is

$\mu(0_{j},\pi)=(-1)^{j-|\pi|}(2!)^{r_{3}}(3!)^{r_{4}}\cdots((j-1)!)^{r_{j}},$

where $r_{i}$ is the number of blocks of $\pi$ of size $i$.

######  Theorem 6.3.

Let $A\in\mathcal{M}_{n}$. Then $A$ is in additive FFP with itself if and only
if $A$ has a single eigenvalue.

###### Proof.

When $A$ has a single eigenvalue then it is in additive FFP with itself by
Corollary 5.11.

Suppose now that $A$ is in additive FFP with itself. By (6.1),
$\kappa_{j}(2A)=2\kappa_{j}(A)$, for $j\geq 1$. On the other hand, by the
definition of a cumulant [2], $\kappa_{j}(2A)=2^{j}\kappa_{j}(A)$. It follows
that

$\kappa_{j}(A)=0,\quad\textrm{ for }j\geq 2.$

It is shown in [2] that for $j\geq 1$,

(6.4)
$m_{j}(A)=\frac{(-1)^{j-1}}{n^{j+1}(j-1)!}\sum_{\pi\in\mathcal{P}(j)}n^{|\pi|}\mu(0_{j},\pi)\kappa_{\pi}(A)\sum_{\rho:\rho\vee\pi=1_{j}}n^{|\rho|}\mu(0_{j},\rho).$

Since $\kappa_{j}(A)=0$, for $j\geq 2$, then in the first summation in (6.4),
all summands vanish, except for the bottom partition $\pi=0_{j}$, and then the
second summation is over the set containing just the top partition
$\rho=1_{j}$, resulting in

$\displaystyle m_{j}(A)$
$\displaystyle=\frac{(-1)^{j-1}}{n^{j+1}(j-1)!}n^{|0_{j}|}\mu(0_{j},0_{j})\kappa_{0_{j}}(A)n^{|1_{j}|}\mu(0_{j},1_{j})$
(6.5)
$\displaystyle=\frac{(-1)^{j-1}}{n^{j+1}(j-1)!}n^{j}\kappa_{1}^{j}(A)n^{1}(-1)^{j-1}(j-1)!$
$\displaystyle=m_{1}^{j}(A).$

The last equation follows from
$\kappa_{1}(A)=m_{1}(A)=\operatorname{\mathrm{tr}}(A)$.

The equations (6), for $j=2,\ldots,n$, imply that $A$ has a single eigenvalue.

∎

##  7\. Matrices in multiplicative finite free position

In addition to additive convolution of polynomials, the notion of
multiplicative convolution was introduced in [10]). We examine here matrices
that are in multiplicative free position and obtain results that are similar
to those in the additive case.

######  Definition 7.1.

Let $p(x)=\sum_{i=0}^{n}a_{i}x^{n-i}$, $q(x)=\sum_{i=0}^{n}b_{i}x^{n-i}$ be
two polynomials of degree $n$ over $\mathbb{C}$. The multiplicative
convolution of $p(x)$ and $q(x)$, denoted $p(x)\boxtimes q(x)$, is

(7.1) $p(x)\boxtimes q(x):=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose
k}}a_{k}b_{k}x^{n-k}.$

The following theorem is the analogue of Theorem 2.3 in the multiplicative
case.

######  Theorem 7.2.

[10] Let $A,B\in\mathcal{M}_{n}$ be normal ($A,B\in\mathbb{R}^{n\times n}$ be
symmetric) matrices. Then

(7.2)
$\chi_{A}(x)\boxtimes\chi_{B}(x)=\int_{\mathcal{U}(n)}\chi_{AU^{*}BU}(x)\,dU=\frac{1}{2^{n}n!}\sum_{P\in\mathcal{P}^{\pm}(n)}\chi_{AP^{T}BP}(x),$

where the expectation is taken over the set of unitary (orthogonal) matrices
$\mathcal{U}(n)$ or the signed permutation matrices $\mathcal{P}^{\pm}(n)$.

######  Definition 7.3.

The matrices $A,B\in\mathcal{M}_{n}$ are in multiplicative finite free
position (or in multiplicative FFP) if

$\chi_{AB}(x)=\chi_{A}(x)\boxtimes\chi_{B}(x).$

The families $\mathcal{E}_{n},\mathcal{F}_{n}\subseteq\mathcal{M}_{n}$ are in
multiplicative finite free position (in multiplicative FFP) if
$\chi_{AB}(x)=\chi_{A}(x)\boxtimes\chi_{B}(x)$ for every $A\in\mathcal{E}_{n}$
and $B\in\mathcal{F}_{n}$.

######  Remarks 7.4.

  1. (1)

As shown in [9], the property of being in multiplicative FFP can be expressed
in terms of mixed discriminants.

  2. (2)

As in the additive case, $\chi_{A}(x)\boxtimes\chi_{B}(x)$ depends on the
characteristic polynomials of the matrices and not on the matrices themselves,
which is not the case for $\chi_{AB}(x)$. However, if $A$ and $B$ are in
multiplicative FFP then so are $PAP^{-1}$ and $PBP^{-1}$, for any regular
matrix $P$.

  3. (3)

For any two matrices $A,B\in\mathcal{M}_{n}$, the leading monomial and the
free coefficient of $\chi_{AB}(x)$ and $\chi_{A}(x)\boxtimes\chi_{B}(x)$ are
the same: $x^{n}$ and $(-1)^{n}\det(AB)=(-1)^{n}\det(A)\det(B)$.

###  7.1. Matrices of dimension $2\times 2$

For matrices $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{{2}\times{2}}$, we have

$\displaystyle\chi_{AB}(x)$
$\displaystyle=x^{2}-(a_{11}b_{11}+a_{12}b_{21}+a_{21}b_{12}+a_{22}b_{22})x$
$\displaystyle+a_{11}a_{22}b_{11}b_{22}+a_{12}a_{21}b_{12}b_{21}-a_{11}a_{22}b_{12}b_{21}-a_{12}a_{21}b_{11}b_{22}$

and

$\displaystyle\chi_{A}(x)\boxtimes\chi_{B}(x)$
$\displaystyle=x^{2}-\frac{1}{2}(a_{11}+a_{22})(b_{11}+b_{22})x$
$\displaystyle+(a_{11}a_{22}-a_{12}a_{21})(b_{11}b_{22}-b_{12}b_{21}).$

By comparing the coefficients of the powers of $x$, we get the following
result.

######  Proposition 7.5.

The matrices $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{{2}\times{2}}$ are in
multiplicative FFP if and only if

$(a_{11}-a_{22})(b_{22}-b_{11})=2(a_{12}b_{21}+a_{21}b_{12}).$

This is the same condition as in (3.1), although here the identical
coefficients are of powers 2 and 0 of $x$, whereas in the additive case the
identical coefficients are of powers 2 an 1 of $x$.

It follows, that also in the multiplicative case, a $2\times 2$ matrix $A$ is
in FFP with itself if and only if it has a single eigenvalue. The analogue of
Proposition 3.2 is the following proposition.

######  Proposition 7.6.

Let $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{{2}\times{2}}$ be in multiplicative
FFP. Then $p(A)$ and $q(B)$ are in multiplicative FFP, for any polynomials
$p(x)$ and $q(x)$. Moreover, if $A$ is regular then $A^{-1}$ and $q(B)$ are in
multiplicative FFP.

###  7.2. Matrices of dimension $3\times 3$

Let us look at the following example.

######  Example 7.7.

Let

$A=\begin{bmatrix}[r]1&0&0\\\ 0&0&0\\\ 0&0&0\end{bmatrix},\qquad
B=\begin{bmatrix}[r]1&1&0\\\ 1&1&0\\\ 0&0&1\end{bmatrix}.$

Then $\chi_{A}(x)=x^{3}-x^{2}$, $\chi_{B}(x)=x^{3}-3x^{2}+2x$,
$\chi_{A+B}(x)=x^{3}-4x^{2}+4x+1$ and $\chi_{AB}(x)=x^{3}-x^{2}$. The additive
and multiplicative convolutions are
$\chi_{A}(x)\boxplus\chi_{B}(x)=x^{3}-4x^{2}+4x-\frac{2}{3}$ and
$\chi_{A}(x)\boxtimes\chi_{B}(x)=x^{3}-x^{2}$. We see that unlike the
situation in dimension $2\times 2$, here the matrices $A$ and $B$ are in
multiplicative FFP but not in additive FFP.

Let us now consider algebraic operations.

######  Example 7.8.

  1.   2. (i)

_Adding a scalar does not preserve multiplicative FFP:_ The matrices $A$ and
$B$ from example 7.7 are in multiplicative FFP, but the pair $(I+A,B)$ is not.

  3. (ii)

_Squaring and inverting do not preserve multiplicative FFP:_ The matrices

$A=\begin{bmatrix}1&0&0\\\ 0&2&0\\\ 0&0&3\end{bmatrix}\qquad
B=\begin{bmatrix}1&-1&2\\\ -1&-2&1\\\ 2&1&1\end{bmatrix}$

are in multiplicative FFP, but the pairs $(A^{2},B)$ and $(A^{-1},B)$ are not.

###  7.3. The lattice of multiplicative finite free varieties

The lattice of multiplicative finite free varieties is constructed in a way
similar to the additive one.

######  Definition 7.9.

Given a family $\mathcal{S}\subseteq\mathcal{M}_{n}$, its multiplicative
finite free complement, denoted ${\mathcal{S}}^{\boxtimes}$, is

${\mathcal{S}}^{\boxtimes}=\\{B\in\mathcal{M}_{n}\,:\,\chi_{AB}(x)=\chi_{A}(x)\boxtimes\chi_{B}(x),\mbox{
for all }A\in\mathcal{S}\\}.$

When ${\mathcal{E}_{n}}^{\boxtimes}=\mathcal{F}_{n}$ and
${\mathcal{F}_{n}}^{\boxtimes}=\mathcal{E}_{n}$, we say that
$\mathcal{E}_{n},\mathcal{F}_{n}$ form a multiplicative finite free
complementary pair (or a multiplicative complementary pair).

######  Proposition 7.10.

If $A$ and $B$ are in multiplicative FFP then so are $\lambda A$ and $B$, for
every $\lambda\in\mathbb{C}$.

###### Proof.

By definition, in both $\chi_{\lambda A}(x)\boxtimes\chi_{B}(x)$ and
$\chi_{\lambda AB}(x)$, the coefficients of $x^{n-k}$ are multiplied by
$\lambda^{k}$ compared to the coefficients of
$\chi_{A}(x)\boxtimes\chi_{B}(x)$ and $\chi_{AB}(x)$. ∎

###  7.4. Diagonal, principally balanced and symmetric matrices

We show here that the pairs of complementary varieties with respect to the
additive convolution, discussed in Section 5, are also complementary pairs in
the multiplicative setting.

######  Theorem 7.11.

Let $\mathcal{D}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
diagonal matrices and let $\mathcal{B}_{n}\subseteq\mathcal{M}_{n}$ be the
family of $n\times n$ principally balanced matrices. Then $\mathcal{D}_{n}$
and $\mathcal{B}_{n}$ form a multiplicative complementary pair.

###### Proof.

The theorem clearly holds for $n=1$, so let us assume that $n\geq 2$. First,
we show that for arbitrary $D\in\mathcal{D}_{n}$ and $B\in\mathcal{B}_{n}$,
the equality $\chi_{DB}(x)=\chi_{D}(x)\boxtimes\chi_{B}(x)$ holds. Let
$D=\mathrm{diag}(d_{1},\ldots,d_{n})$,

$\chi_{D}(x)=\sum_{i=0}^{n}a_{i}x^{n-i}=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}d_{J}\bigg{)}x^{n-i},$

where $d_{J}=d_{j_{1}}\cdots d_{j_{i}}$ for $J=\\{j_{1},\ldots,j_{i}\\}$. Let
also

$\chi_{B}(x)=\sum_{i=0}^{n}b_{i}x^{n-i}=\sum_{i=0}^{n}(-1)^{i}{n\choose
i}m_{i}x^{n-i},$

where $m_{i}$, $i=0,\ldots,n$, is the value of a principal minor of $B$ of
order $i$. Since

$DB=\begin{bmatrix}d_{1}B_{1\,.}\\\ d_{2}B_{2\,.}\\\ \vdots\\\
d_{n}B_{n\,.}\end{bmatrix},$

where $B_{i\,.}$ denotes the $i$-th row of $B$, we have

$\displaystyle\chi_{DB}(x)$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}\det((DB)_{J})\bigg{)}x^{n-i}$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}d_{J}\det(B_{J})\bigg{)}x^{n-i}$
$\displaystyle=\sum_{i=0}^{n}\bigg{(}(-1)^{i}\sum_{J\subseteq[n],\,|J|=i}d_{J}m_{i}\bigg{)}x^{n-i}$
(7.3)
$\displaystyle=\sum_{i=0}^{n}a_{i}m_{i}x^{n-i}=\sum_{i=0}^{n}a_{i}\frac{(-1)^{i}b_{i}}{{n\choose
i}}x^{n-i}.$

By Definition 7.1, this is exactly $\chi_{D}(x)\boxtimes\chi_{B}(x)$ and it
follows that $\mathcal{B}_{n}\subseteq{\mathcal{D}_{n}}^{\boxtimes}$ and
$\mathcal{D}_{n}\subseteq{\mathcal{B}_{n}}^{\boxtimes}$

Next, we show that ${\mathcal{D}_{n}}^{\boxtimes}\subseteq\mathcal{B}_{n}$.
Let $B=(b_{ij})\in{\mathcal{D}_{n}}^{\boxtimes}$. Let $K\subseteq[n]$,
$|K|=k$, $0<k<n$, and let $D(\lambda,K)=(d_{ij})$ be the parametric diagonal
matrix with entries

$d_{ij}=\left\\{\begin{array}[]{rcl}0&\mbox{for}&i\neq j,\\\
\lambda&\mbox{for}&i=j\in K,\\\ 1&\mbox{for}&i=j\notin K.\end{array}\right.$
$\displaystyle\chi_{D(\lambda,K)B}(x)$
$\displaystyle=\det(xI-(D(\lambda,K)B))$
$\displaystyle=x^{n}+p_{1}(\lambda)x^{n-1}+\cdots+p_{n-1}(\lambda)x+(-1)^{n}\det(D(\lambda,K)B),$

where $p_{j}(\lambda)$ is a polynomial in $\lambda$ of degree at most
$\max(j,k)$. Let $p^{(i)}(x)$ be the $i$-th derivative of $p(x)$ with respect
to $x$. Then

$\displaystyle(\chi_{D(\lambda,K)B})^{(n-k)}(0)$
$\displaystyle=(n-k)!p_{k}(\lambda)$
$\displaystyle=(-1)^{k}(n-k)!\lambda^{k}\det(B_{K})+q(\lambda),$

where $q(\lambda)$ is a polynomial in $\lambda$ of degree less than $k$. We
have

(7.4)
$\lim_{\lambda\to\infty}\frac{(\chi_{D(\lambda,K)B})^{(n-k)}(0)}{(-1^{k})(n-k)!\lambda^{k}}=\det(B_{K}).$

We now compute the same limit for
$\chi_{D(\lambda,K)}(x)\boxtimes\chi_{B}(x)$. The characteristic polynomial of
$D(\lambda,K)(x)$ is

$\chi_{D(\lambda,K)}(x)=\sum_{i=0}^{n}a_{i}x^{n-i}=(x-\lambda)^{k}(x-1)^{n-k}.$

Then,

(7.5) $a_{k}=(-1)^{k}\lambda^{k}+g(\lambda),$

where $g(\lambda)$ is a polynomial in $\lambda$ of degree less than $k$. The
characteristic polynomial of $B$ is $\chi_{B}(x)=\sum_{i=0}^{n}b_{i}x^{n-i}$
with

(7.6) $b_{k}=(-1)^{k}\sum_{I\subseteq[n],\,|I|=k}\det(B_{I}).$

By (7.1),

$\displaystyle(\chi_{D(\lambda,K)}\boxtimes\chi_{B})^{(n-k)}(0)$
$\displaystyle=\left(\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose
k}}a_{k}b_{k}x^{n-k}\right)^{(n-k)}(0)$
$\displaystyle=\frac{(-1)^{k}(n-k)!}{{n\choose k}}a_{k}b_{k}$

Substituting for $a_{k}$ and $b_{k}$ the expressions in (7.5) and (7.6), we
get

$=\frac{(-1)^{k}(n-k)!}{{n\choose
k}}((-1)^{k}\lambda^{k}+g(\lambda))(-1)^{k}\sum_{I\subseteq[n],\,|I|=k}\det(B_{I}).$

Computing the limit as in (7.4) gives

(7.7)
$\lim_{\lambda\to\infty}\frac{(\chi_{D(\lambda,K)}\boxtimes\chi_{B})^{(n-k)}(0)}{(-1^{k})(n-k)!\lambda^{k}}=\frac{1}{{n\choose
k}}\sum_{I\subseteq[n],\,|I|=k}\det(B_{I}).$

Comparing (7.4) with (7.7), we get that

$\det(B_{K})=\frac{1}{{n\choose k}}\sum_{I\subseteq[n],\,|I|=k}\det(B_{I}),$

and as in Theorem 5.7, it follows that $B$ is principally balanced and
therefore ${\mathcal{D}_{n}}^{\boxtimes}\subseteq\mathcal{B}_{n}$. Together
with the inclusion in the opposite direction, we have
${\mathcal{D}_{n}}^{\boxtimes}=\mathcal{B}_{n}$.

Finally, we show that ${\mathcal{B}_{n}}^{\boxtimes}\subseteq\mathcal{D}_{n}$.
So, assume by contradiction that $A=(a_{ij})\in{\mathcal{B}_{n}}^{\boxtimes}$
and $A\notin\mathcal{D}_{n}$. It follows that for some $(j_{0},i_{0})$,
$j_{0}\neq i_{0}$, $a_{j_{0}i_{0}}=a\neq 0$. Let $B=(b_{ij)})$ be the matrix
$B=I_{n}+E_{n}(i_{0}j_{0})$:

$b_{ij}=\left\\{\begin{array}[]{rl}1&\mbox{for }i=j,\\\ 1&\mbox{for
}(i,j)=(i_{0},j_{0}),\\\ 0&\mbox{otherwise.}\end{array}\right.$

Clearly, $B\in\mathcal{B}_{n}$. Let the characteristic polynomial of $A$ be
$\chi_{A}(x)=\sum_{i=0}^{n}a_{i}x^{n-i}$ and that of $B$:

$\chi_{B}(x)=\sum_{i=0}^{n}b_{i}x^{n-i}=(x-1)^{n}=\sum_{i=0}^{n}{n\choose
i}(-i)^{i}x^{n-i}.$

Since $\operatorname{\mathrm{Tr}}(AB)=\operatorname{\mathrm{Tr}}(A)+a$, the
characteristic polynomial of $AB$ is

(7.8) $\chi_{AB}(x)=x^{n}+(a_{1}-a)x^{n-1}+q(x),$

where $q(x)$ is a polynomial in $x$ of degree less than $n-1$.

The multiplicative convolution of $\chi_{A}(x)$ and $\chi_{B}(x)$ is

(7.9) $\displaystyle\chi_{A}(x)\boxtimes\chi_{B}(x)$
$\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose k}}a_{k}b_{k}x^{n-k}$
(7.10) $\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose
k}}a_{k}{n\choose k}(-1)^{k}x^{n-k}$ (7.11)
$\displaystyle=\sum_{k=0}^{n}a_{k}x^{n-k}=\chi_{A}(x).$

We see that the coefficients of $x^{n-1}$ in (7.8) and in (7.9) are different
from each other, implying that
$\chi_{AB}(x)\neq\chi_{A}(x)\boxtimes\chi_{B}(x)$. It follows that
${\mathcal{B}_{n}}^{\boxtimes}\subseteq\mathcal{D}_{n}$ and with the inclusion
in the opposite direction, we have
${\mathcal{B}_{n}}^{\boxtimes}=\mathcal{D}_{n}$. ∎

The next corollary is analogues to Corollary 5.8 and is proved in a similar
way.

######  Corollary 7.12.

Let $\mathcal{A}_{n}\subseteq\mathcal{M}_{n}$ be a maximal family of commuting
diagonalizable matrices and let $\mathcal{B}_{n}$ be the family of principally
balanced matrices. Then there exists a non-singular matrix $P$, such that
$\mathcal{A}_{n}$ and $P^{-1}{\mathcal{B}_{n}}P$ form a multiplicative
complementary pair.
If $\mathcal{A}_{n}$ is a maximal family of commuting symmetric matrices then
there exists an orthogonal matrix $U$, such that $\mathcal{A}_{n}$ and
$U^{T}{\mathcal{B}_{n}}U$ form a multiplicative complementary pair.

Similarly to Theorem 2.3 we have the following theorem of [10] with respect to
the multiplicative convolution.

######  Theorem 7.13.

Let $A,B\in\mathbb{R}^{n\times n}$ with $A$ being symmetric. Then

(7.12) $\chi_{A}(x)\boxtimes\chi_{B}(x)=\mathbb{E}_{U}\,[\chi_{AU^{T}BU}(x)],$

where the expectation is taken over the set of orthogonal matrices $U$.

###  7.5. Triangular matrices

As with the additive case, the upper triangular matrices and the upper
triangular matrices with constant diagonal form a multiplicative complementary
pair, and similarly for lower triangular matrices.

######  Theorem 7.14.

Let $\mathcal{R}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
upper triangular matrices and let
$\widehat{\mathcal{R}}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times
n$ upper triangular matrices with constant diagonal. Then $\mathcal{R}_{n}$
and $\widehat{\mathcal{R}}_{n}$ form a multiplicative complementary pair.

###### Proof.

The lines of proof are analogous to those of Theorem 5.10. We only mention
that when $A=(a_{ij})\notin\mathcal{R}_{n}$ with $a_{j_{0}i_{0}}=a\neq 0$, for
some $j_{0}>i_{0}$, then we define $B=(b_{ij})\in\widehat{\mathcal{R}}_{n}$ to
be the matrix $B=I_{n}+E_{n}(i_{0}j_{0})$ as in Theorem 7.11. Then
$\chi_{A}(x)\boxtimes\chi_{B}(x)=\chi_{A}(x)$ whereas
$\chi_{AB}(x)\neq\chi_{A}(x)$. ∎

###  7.6. Scalar matrices

Also for scalar matrices the multiplicative convolution behaves analogously to
the additive convolution.

######  Theorem 7.15.

Let $\mathcal{S}_{n}\subseteq\mathcal{M}_{n}$ be the family of $n\times n$
scalar matrices. Then $\mathcal{S}_{n}$ and $\mathcal{M}_{n}$ form a
multiplicative complementary pair.

###### Proof.

First, we show that when $A=(a_{ij})\in\mathcal{M}_{n}$ and
$C=\mathrm{diag}(c,\ldots,c)$ then
$\chi_{AC}(x)=\chi_{A}(x)\boxtimes\chi_{C}(x)$. As before, let

$\chi_{A}(x)=\sum_{i=0}^{n}a_{i}x^{n-i},$

and

$\chi_{C}(x)=\sum_{i=0}^{n}c_{i}x^{n-i}=\sum_{i=0}^{n}(-1)^{i}{n\choose
i}c^{i}x^{n-i}.$

Then

$\chi_{AC}(x)=\sum_{k=0}^{n}a_{k}c^{k}x^{n-k}$

and

$\displaystyle\chi_{A}(x)\boxtimes\chi_{C}(x)$
$\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose k}}a_{k}c_{k}x^{n-k}$
$\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose
k}}a_{k}(-1)^{k}{n\choose k}c^{k}x^{n-k}$
$\displaystyle=\sum_{k=0}^{n}a_{k}c^{k}x^{n-k}=\chi_{AC}(x).$

The rest of the proof is similar to that of Theorem 5.9. ∎

###  7.7. Matrices in multiplicative FFP with themselves

The following result is similar to what happens in the additive case.

######  Proposition 7.16.

Let $A\in\mathcal{M}_{n}$. Then $A$ is in multiplicative FFP with itself if
and only if $A$ has a single eigenvalue.

###### Proof.

By similarity transformation, we can assume, without loss of generality, that
$A$ is in upper triangular form with its eigenvalues
$\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ on the diagonal. Let
$\chi_{A}(x)=\sum_{k=0}^{n}a_{k}x^{n-k}$, then

$a_{k}=(-1)^{k}\sum_{\begin{subarray}{c}I=\\{i_{1},i_{2},\ldots,i_{k}\\}\\\
1\leq i_{1}<i_{2}<\cdots<i_{k}\leq
n\end{subarray}}\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{k}}.$

Hence

(7.13)
$\chi_{A^{2}}(x)=\sum_{k=0}^{n}(-1)^{k}\sum_{\begin{subarray}{c}I=\\{i_{1},i_{2},\ldots,i_{k}\\}\\\
1\leq i_{1}<i_{2}<\cdots<i_{k}\leq
n\end{subarray}}\lambda^{2}_{i_{1}}\lambda^{2}_{i_{2}}\cdots\lambda^{2}_{i_{k}}\,x^{n-k}.$

On the other hand,

$\displaystyle\chi_{A}(x)\boxtimes\chi_{A}(x)$
$\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose k}}a^{2}_{k}x^{n-k}$
(7.14) $\displaystyle=\sum_{k=0}^{n}\frac{(-1)^{k}}{{n\choose
k}}\Big{(}\sum_{\begin{subarray}{c}I=\\{i_{1},i_{2},\ldots,i_{k}\\}\\\ 1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq
n\end{subarray}}\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{k}}\Big{)}^{2}x^{n-k}.$

By comparing the corresponding coefficients in (7.13) and (7.7), for
$k=1,\ldots,n-1$, we get that $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{n}$.

In the other direction, when all eigenvalues of $A$ are the same then $A$ is
similar to an upper triangular matrix with a constant diagonal and by Theorem
7.14, $A$ is in multiplicative FFP with itself. ∎

###  7.8. Moments of products of matrices that are in multiplicative FFP

Moments and cumulants for the multiplicative convolution of polynomials were
handled by Arizmendi et al. [1]. Among others, they derived a formula for the
finite free cumulants of the multiplicative convolution of polynomials $p$ and
$q$ in terms of the finite free cumulants of $p$ and of $q$. They also were
able to express the moments of the empirical root distribution of $p\boxtimes
q$ in terms of the finite free cumulants of $p$ and the moments of $q$. As
shown in [1], finite free multiplicative convolution converges to the free
multiplicative convolution.

As in the additive case, when $A$ and $B$ are in multiplicative FFP, we can
compare the coefficients of $\chi_{A}(x)\boxtimes\chi_{B}(x)$ with the
corresponding coefficients of $\chi_{AB}(x)$ and derive formulas for the
moments of $AB$ in terms of those of $A$ and $B$. Next, we derive these
formulas for the first and second moments.

######  Proposition 7.17.

Let $A,B\in\mathcal{M}_{n}$ be in multiplicative FFP. Then

  1. (1)

$m_{1}(AB)=m_{1}(A)m_{1}(B)$.

  2. (2)

$m_{2}(AB)=\frac{n}{n-1}[m_{2}(A)m_{1}^{2}(B)+m_{1}^{2}(A)m_{2}(B)-m_{1}^{2}(A)m_{1}^{2}(B)]-\frac{1}{n-1}m_{2}(A)m_{2}(B).$

###### Proof.

(1) Let $\chi_{A}(x)=\sum_{k=0}^{n}a_{k}x^{n-k}$ and
$\chi_{B}(x)=\sum_{k=0}^{n}b_{k}x^{n-k}$. The coefficient of $x^{n-1}$ in
$\chi_{A}(x)\boxtimes\chi_{B}(x)$ is

(7.15)
$\displaystyle-\frac{1}{n}a_{1}b_{1}=-\frac{1}{n}\operatorname{\mathrm{Tr}}(A)\operatorname{\mathrm{Tr}}(B)=-n\cdot
m_{1}(A)m_{1}(B),$

whereas the coefficient of $x^{n-1}$ in $\chi_{AB}(x)$ is

(7.16) $-\operatorname{\mathrm{Tr}}(AB)=-n\cdot m_{1}(AB).$

The result then follows.

(2) The coefficient of $x^{n-2}$ in $\chi_{A}(x)\boxtimes\chi_{B}(x)$ is
$-\frac{1}{{n\choose 2}}a_{2}b_{2}$, which equals, by (6.2),

$-\frac{2}{n(n-1)}\left(\frac{\operatorname{\mathrm{Tr}}^{2}(A)}{2}-\frac{\operatorname{\mathrm{Tr}}(A^{2})}{2}\right)\left(\frac{\operatorname{\mathrm{Tr}}^{2}(B)}{2}-\frac{\operatorname{\mathrm{Tr}}(B^{2})}{2}\right).$

In terms of moments it equals

$\displaystyle\frac{n}{2(n-1)}[$ $\displaystyle n^{2}\cdot
m_{1}^{2}(A)m_{1}^{2}(B)+m_{2}(A)m_{2}(B)$ (7.17) $\displaystyle-n\cdot
m_{1}^{2}(A)m_{2}(B)-n\cdot m_{2}(A)m_{1}^{2}(B)].$

The coefficient of $x^{n-2}$ in $\chi_{AB}(x)$ is

$\frac{\operatorname{\mathrm{Tr}}^{2}(AB)}{2}-\frac{\operatorname{\mathrm{Tr}}(AB)^{2}}{2}.$

In terms of moments and by formula (1) it equals

(7.18) $\frac{n^{2}}{2}\cdot m_{1}^{2}(A)m_{1}^{2}(B)-\frac{n}{2}\cdot
m_{2}(AB).$

Formula (2) is then obtained by the fact that (7.8) and (7.18) are equal for
matrices in multiplicative FFP. ∎

Comparing other coefficients of powers of $x$ do not result in simple and nice
formulas for higher moments of $AB$.

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