Generalization of Roth's solvability criteria to systems of matrix equations

TL;DR

This paper extends Roth's classical solvability criterion for matrix equations to complex and quaternion matrix systems, providing new conditions for the existence of solutions in these broader contexts.

Contribution

The paper generalizes Roth's criterion to systems involving complex conjugation and quaternion matrices, broadening the applicability of solvability conditions for matrix equations.

Findings

Extended Roth's criterion to complex conjugate matrix systems

Established solvability conditions for quaternion matrix equations

Unified framework for real, complex, and quaternion matrix equations

Abstract

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{\sigma_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^{\sigma}$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.

Equations

$$A_i{X}_{i^{\prime }}{M}_i-N_i{X}_{i^{\prime \prime }}^{{\sigma }_i}{B}_i=C_i,i=1,\dots ,s$$

$$A_i{X}_{i^{\prime }}^{{\epsilon }_i}{M}_i-N_i{X}_{i^{\prime \prime }}^{{\delta }_i}{B}_i=C_i,i^{\prime },i^{\prime \prime }\in \{1,\dots ,t\},i=1,\dots ,s$$

$$(a+b{)}^{\complement }=a^{\complement }+b^{\complement },(ab{)}^{\complement }=a^{\complement }{b}^{\complement },(a^{\complement }{)}^{\complement }=a\text{for all a∈F.}$$

$$(a+b{)}^{\circ }=a^{\circ }+b^{\circ },(ab{)}^{\circ }=b^{\circ }{a}^{\circ },(a^{\circ }{)}^{\circ }=a\text{for all a∈F.}$$

$$A_i{X}_{i^{\prime }}^{{\epsilon }_i}-X_{i^{\prime \prime }}^{{\delta }_i}{B}_i=C_i,i=1,\dots ,s$$

$$A^{†}:=(A^{\circ }{)}^{\top },A^{\text{Asterisk}}:=((A^{\complement }{)}^{\circ }{)}^{\top }$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]P_{i^{\prime }}^{⟨{\epsilon }_i⟩}=P_{i^{\prime \prime }}^{⟨{\delta }_i⟩}\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right],i=1,\dots ,s,$$

$$P^{⟨\sigma ⟩}:=\{\begin{array}{ll}{P}^{\sigma }& \text{if }\sigma \in \{1,\complement \},\\ J(P^{\sigma }{)}^{-1}{J}^{-1}& \text{if }\sigma \in \{†,\text{Asterisk}\},\end{array}J:=\left[\begin{array}{cc}0& I\\ -I& 0\end{array}\right].$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]P_{i^{\prime }}^{{\epsilon }_i}$$

$$P_{i^{\prime \prime }}^{{\delta }_i}\left[\begin{array}{cc}0& -B_i\\ A_i& 0\end{array}\right]P_{i^{\prime }}^{{\epsilon }_i}$$

$$\left[\begin{array}{cc}0& -A_i\\ B_i& 0\end{array}\right]$$

$$P_{i^{\prime \prime }}^{{\delta }_i}\left[\begin{array}{cc}{B}_i& 0\\ 0& A_i\end{array}\right]$$

$$h^{\complement }:=a+bi-cj-dk,h^{\circ }:=a-bi+cj+dk,\bar{h}=(h^{\complement }{)}^{\circ }=a-bi-cj-dk$$

$$A^{†}=(A^{\circ }{)}^{\top },A^{\text{Asterisk}}={\bar{A}}^{\top }$$

$$\begin{array}{rl}\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]Q_i& =R_i\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right]\\ \left[\begin{array}{cc}I& 0\\ 0& M_i\end{array}\right]Q_i& =P_{i^{\prime }}^{⟨{\epsilon }_i⟩}\left[\begin{array}{cc}I& 0\\ 0& M_i\end{array}\right]\\ \left[\begin{array}{cc}{N}_i& 0\\ 0& I\end{array}\right]P_{i^{\prime \prime }}^{⟨{\delta }_i⟩}& =R_i\left[\begin{array}{cc}{N}_i& 0\\ 0& I\end{array}\right]\end{array}\},i=1,\dots ,s.$$

$$\begin{array}{rl}{A}_i{Y}_i-Z_i{B}_i& =C_i\\ Y_i-X_{i^{\prime }}^{{\epsilon }_i}{M}_i& =0\\ N_i{X}_{i^{\prime \prime }}^{{\delta }_i}-Z_i& =0\end{array}\},i=1,\dots ,s$$

$$A_i{X}_{i^{\prime }}^{{\alpha }_i}-X_{i^{\prime \prime }}^{{\beta }_i}{B}_i=C_i,i=1,\dots ,s$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]P_{i^{\prime }}^{{\alpha }_i}=P_{i^{\prime \prime }}^{{\beta }_i}\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right],i=1,\dots ,s.$$

$$P_1=\left[\begin{array}{cc}I& {\underline{X}}_1\\ 0& I\end{array}\right],\dots ,P_t=\left[\begin{array}{cc}I& {\underline{X}}_t\\ 0& I\end{array}\right].$$

$$(P_1,\dots ,P_t)\in \mathcal{U}:={\mathbb{F}}^{n_1\times n_1}\oplus \cdots \oplus {\mathbb{F}}^{n_t\times n_t}.$$

$${\mathcal{U}}_1$$

$${\mathcal{U}}_2$$

$$U=\left(\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{13}& U_{14}\end{array}\right],\dots ,\left[\begin{array}{cc}{U}_{t1}& U_{t2}\\ U_{t3}& U_{t4}\end{array}\right]\right)\in \mathcal{U}$$

$${\pi }_k:\left(\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{13}& U_{14}\end{array}\right],\dots ,\left[\begin{array}{cc}{U}_{t1}& U_{t2}\\ U_{t3}& U_{t4}\end{array}\right]\right)\mapsto \left(\left[\begin{array}{cc}{U}_{11}& 0\\ U_{13}& 0\end{array}\right],\dots ,\left[\begin{array}{cc}{U}_{t1}& 0\\ U_{t3}& 0\end{array}\right]\right).$$

$${\dim }_{\mathbb{G}}\mathrm{Im}{\pi }_k+{\dim }_{\mathbb{G}}\mathrm{Ker}{\pi }_k={\dim }_{\mathbb{G}}{\mathcal{U}}_k,k=1,2.$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right](U_{i^{\prime }}{P}_{i^{\prime }}{)}^{{\alpha }_i}=U_{i^{\prime \prime }}^{{\beta }_i}\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]P_{i^{\prime }}^{{\alpha }_i}=(U_{i^{\prime \prime }}{P}_{i^{\prime \prime }}{)}^{{\beta }_i}\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right].$$

$$U=\left(\left[\begin{array}{cc}0& U_{12}\\ 0& U_{14}\end{array}\right],\dots ,\left[\begin{array}{cc}0& U_{t2}\\ 0& U_{t4}\end{array}\right]\right)\in {\mathcal{U}}_1$$

$$U=\left(\left[\begin{array}{cc}0& U_{12}\\ 0& U_{14}\end{array}\right],\dots ,\left[\begin{array}{cc}0& U_{t2}\\ 0& U_{t4}\end{array}\right]\right)\in {\mathcal{U}}_2$$

$$U=\left(\left[\begin{array}{cc}{U}_{11}& 0\\ U_{13}& 0\end{array}\right],\dots ,\left[\begin{array}{cc}{U}_{t1}& 0\\ U_{t3}& 0\end{array}\right]\right)\in \mathrm{Im}{\pi }_1,$$

$$\left(\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{13}& U_{14}\end{array}\right],\dots ,\left[\begin{array}{cc}{U}_{t1}& U_{t2}\\ U_{t3}& U_{t4}\end{array}\right]\right)\in {\mathcal{U}}_1,$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]\left[\begin{array}{cc}{U}_{i^{\prime }1}^{{\alpha }_i}& U_{i^{\prime }2}^{{\alpha }_i}\\ U_{i^{\prime }3}^{{\alpha }_i}& U_{i^{\prime }4}^{{\alpha }_i}\end{array}\right]=\left[\begin{array}{cc}{U}_{i^{\prime \prime }1}^{{\beta }_i}& U_{i^{\prime \prime }2}^{{\beta }_i}\\ U_{i^{\prime \prime }3}^{{\beta }_i}& U_{i^{\prime \prime }4}^{{\beta }_i}\end{array}\right]\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right],i=1,\dots ,s.$$

$$\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right]\left[\begin{array}{cc}{U}_{i^{\prime }1}^{{\alpha }_i}& 0\\ U_{i^{\prime }3}^{{\alpha }_i}& 0\end{array}\right]=\left[\begin{array}{cc}{U}_{i^{\prime \prime }1}^{{\beta }_i}& 0\\ U_{i^{\prime \prime }3}^{{\beta }_i}& 0\end{array}\right]\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right],i=1,\dots ,s,$$

Full text

AMS classification: 15A24

Keywords: Systems of matrix equations, Sylvester equations, Roth’s criteria

1 Introduction

Roth [13] proved that the matrix equation $AX-XB=C$ (respectively, $AX-YB=C$) over a field has a solution if and only if the matrices $\left[\begin{smallmatrix}A&C\\ 0&B\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}A&0\\ 0&B\end{smallmatrix}\right]$ are similar (respectively, equivalent); see also [8, Section 4.4.22] and [10, Section 12.5].

Dmytryshyn and Kågström [4, Theorem 6.1] extended Roth’s criteria to the system of generalized Sylvester equations

[TABLE]

with unknown matrices $X_{1},\dots,X_{t}$ over a field of characteristic not 2 with a fixed involution, in which every $X_{i^{\prime\prime}}^{\sigma_{i}}$ is either $X_{i^{\prime\prime}}$, or $X_{i^{\prime\prime}}^{\top}$, or $X^{*}_{i^{\prime\prime}}$. Most of the known generalizations of Roth’s criteria are special cases of their criterion. The first author was awarded the SIAM Student Paper Prize 2015 for the paper [4].

However, Dmytryshyn and Kågström [4] do not consider complex matrix equations that include the complex conjugate of unknown matrices. The theory of such equations and their applications to discrete-time antilinear systems are presented in Wu and Zhang’s new book [17]. Bevis, Hall, and Hartwig [1] proved that the complex matrix equation $A\bar{X}-XB=C$ has a solution if and only if the matrices $\left[\begin{smallmatrix}A&C\\ 0&B\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}A&0\\ 0&B\end{smallmatrix}\right]$ are consimilar (i.e., $\bar{S}^{-1}\left[\begin{smallmatrix}A&C\\ 0&B\end{smallmatrix}\right]S=\left[\begin{smallmatrix}A&0\\ 0&B\end{smallmatrix}\right]$ for some nonsingular $S$).

We extend Dmytryshyn and Kågström’s criterion to a large class of matrix equations that includes the systems

[TABLE]

• •

of complex matrix equations, in which $\varepsilon_{i},\delta_{i}\in\{1,\complement,\top,\Asterisk\}$, where $X^{\complement}:=\bar{X}$ is the complex conjugate matrix and $X^{\Asterisk}:=\bar{X}^{\top}$ is the complex adjoint matrix, and

• •

of quaternion matrix equations, in which $\varepsilon_{i},\delta_{i}\in\{1,\Asterisk\}$, where $X^{\Asterisk}$ is the quaternion adjoint matrix.

We prove our criterion by methods of [4] (see also [6, 15, 16]), though our exposition is self-contained and uses only elementary linear algebra.

Note that the system of matrix equations (1) over a field can be rewritten as a system $Mx=b$ of linear equations, which gives another criterion of solvability for (1): it has a solution if and only if $\operatorname{rank}M=\operatorname{rank}[M|b]$. However, the system $Mx=b$ is large and can be ill-conditioned.

Special cases of the system (1) are considered in hundreds of articles and books. For recent results related to solvability criteria we refer the reader to [2, 3, 4, 5, 7, 14, 17] and the references given there. A survey of papers on Roth’s criteria and their generalizations is given in the extended introduction to [7]. A quaternion linear algebra is presented in [12], in which quaternion matrix equations are considered in Chapters 5 and 14.

2 Main results

Let $\mathbb{F}$ be a skew field (which can be a field). An involutory automorphism of $\mathbb{F}$ is a bijection $a\mapsto a^{\complement}$ of $\mathbb{F}$ onto itself, satisfying

[TABLE]

An involutory anti-automorphism of $\mathbb{F}$ is a bijection $a\mapsto a^{\circ}$, satisfying

[TABLE]

For example, the complex conjugation is an involutory automorphism and involutory anti-automorphism of $\mathbb{C}$; the quaternion conjugation is an involutory anti-automorphism of $\mathbb{H}$.

The following theorem is proved in Section 3.

Theorem 1**.**

Given

• •

a skew field $\mathbb{F}$ of characteristic not 2 that is finite dimensional over its center,

• •

an involutory automorphism $a\mapsto a^{\complement}$ $($possible, the identity$)$ and an involutory anti-automorphism $a\mapsto a^{\circ}$ of $\mathbb{F}$ $($possible, the identity if $\mathbb{F}$ is a field$)$,

• •

a system

[TABLE]

of matrix equations over $\mathbb{F}$ with unknown matrices $X_{1},\dots,X_{t}$, in which all $i^{\prime},i^{\prime\prime}\in\{1,\dots,t\}$, $\varepsilon_{i},\delta_{i}\in\{1,\complement,\dagger,\Asterisk\}$, and

[TABLE]

for each matrix $A$ over $\mathbb{F}$;

the system (2) has a solution if and only if there exist nonsingular matrices $P_{1},\dots,P_{t}$ over $\mathbb{F}$ such that

[TABLE]

in which

[TABLE]

If all $\varepsilon_{i},\delta_{i}\in\{1,\complement\}$ in (2), then the condition “$\mathbb{F}$ of characteristic not 2” in Theorem 1 can be omitted; see Lemma 1.

The conditions (3) on the block matrices from Theorem 1 are all given in the same style using (4). In the following remark, we give these conditions more explicitly for each of four possible cases.

Remark 1*.*

For each $i=1,\dots,s$, the equality (3) in Theorem 1 can be rewritten in the form:

[TABLE]

Corollary 1**.**

• (a)

Over $\mathbb{R}$, the system (2) with $\varepsilon_{i},\delta_{i}\in\{1,\top\}$ has a solution if and only if (3) holds for some nonsingular real matrices $P_{1},\dots,P_{t}$, and $\top$ is used instead of $\dagger$ in (4).

• (b)

Over $\mathbb{C}$, the system (2) with $\varepsilon_{i},\delta_{i}\in\{1,\complement,\top,\Asterisk\}$ has a solution if and only if (3) holds for some nonsingular complex matrices $P_{1},\dots,P_{t}$. Here $A^{\complement}:=\bar{A}$ is the complex conjugate matrix, $A^{\Asterisk}:=\bar{A}^{\top}$ is the complex adjoint matrix. The symbol $\top$ is used instead of $\dagger$ in (4).

• (c)

Over $\mathbb{H}$, the system (2) with $\varepsilon_{i},\delta_{i}\in\{1,\complement,\dagger,\Asterisk\}$ has a solution if and only if (3) holds for some nonsingular quaternion matrices $P_{1},\dots,P_{t}$. Here

[TABLE]

for each quaternion $h=a+bi+cj+dk$, and

[TABLE]

for each quaternion matrix $A$.

Note that each involutory automorphism of $\mathbb{H}$ is either the identity, or $h\mapsto a+bi-cj-dk$ in a suitable set of orthogonal imaginary units $i,j,k\in\mathbb{H}$, see [9, Lemma 1]; and each involutory anti-automorphism of $\mathbb{H}$ is either $h\mapsto a-bi+cj+dk$, or $h\mapsto a-bi-cj-dk$ in a suitable set of orthogonal imaginary units, see [12, Theorem 2.4.4(c)].

Theorem 2**.**

Let $\mathbb{F}$ be a skew field of characteristic not $2$ that is finite dimensional over its center. The system (1) over $\mathbb{F}$, in which all $\varepsilon_{i}$ and $\delta_{i}$ are as in Theorem 1, has a solution if and only if there exist nonsingular matrices $P_{1},\dots,P_{t},$​ $Q_{1},\dots,Q_{s},$​ $R_{1},\dots,R_{s}$ over $\mathbb{F}$ satisfying the following $3s$ equalities:

[TABLE]

Proof (assuming that Theorem 1 holds).

Define from (1) the system of $3s$ matrix equations

[TABLE]

with unknown matrices $X_{1},\dots,X_{t},\,Y_{1},\dots,Y_{s},\,Z_{1},\dots,Z_{s}$. If the system (1) has a solution $(\underline{X}_{1},\dots,\underline{X}_{t})$, then (6) has the solution $(\underline{X}_{1},\dots,\underline{X}_{t};\underline{Y}_{1},\dots,\underline{Y}_{s};\underline{Z}_{1},\dots,\underline{Z}_{s})$, in which all $\underline{Y}_{i}:=\underline{X}_{i^{\prime}}^{\varepsilon_{i}}M_{i}$ and $\underline{Z}_{i}:=N_{i}\underline{X}_{i^{\prime\prime}}^{\delta_{i}}$. Thus, the system (1) has a solution if and only if (6) has a solution. By Theorem 1, the system (6) has a solution if and only if (5) holds for some nonsingular matrices $P_{1},\dots,P_{t},Q_{1},\dots,Q_{s},R_{1},\dots,R_{s}$. ∎

3 The proof of Theorem 1

The following lemma proves Theorem 1 if all $\varepsilon_{i},\delta_{i}\in\{1,\complement\}$.

Lemma 1**.**

Let $\mathbb{F}$ be a skew field that is finite dimensional over its center. Let $a\mapsto a^{\complement}$ be an involutory automorphism of $\mathbb{F}$ $($which can be the identity$)$. Let

[TABLE]

be a system of matrix equations over $\mathbb{F}$ with unknown matrices $X_{1},\dots,X_{t}$, in which all $\alpha_{i},\beta_{i}\in\{1,\complement\}$. Then the system (7) has a solution if and only if there exist nonsingular matrices $P_{1},\dots,P_{t}$ such that

[TABLE]

Proof.

$\Longrightarrow$. If $(\underline{X}_{1},\dots,\underline{X}_{t})$ is a solution of (7), then (8) holds for

[TABLE]

$\Longleftarrow$. Suppose there are nonsingular matrices $P_{1},\dots,P_{t}$ of sizes ${n_{1}\times n_{1}},\dots,n_{t}\times n_{t}$ satisfying (8). Then

[TABLE]

Denote by $C(\mathbb{F})$ the center of $\mathbb{F}$ (which coincides with $\mathbb{F}$ if $\mathbb{F}$ is a field). For $c\in C(\mathbb{F})$ and any $a\in\mathbb{F}$, $a^{\complement}c^{\complement}=(ac)^{\complement}=(ca)^{\complement}=c^{\complement}a^{\complement}$, and so $c^{\complement}\in C(\mathbb{F})$. Hence $c\mapsto c^{\complement}$ is an automorphism of $C(\mathbb{F})$ of order 1 or 2. By [11, Chapter VI, Theorem 1.8], the index of the subfield $\mathbb{G}:=\{c\in C(\mathbb{F})\,|\,c=c^{\complement}\}$ in $C(\mathbb{F})$ is 1 or 2. Since $\mathbb{F}$ is finite dimensional over its center, $\mathbb{F}$ is also finite dimensional over $\mathbb{G}$.

Thus, the set $\mathcal{U}$ in (10) is a finite dimensional vector space over $\mathbb{G}$. Define its subspaces

[TABLE]

Let the matrices of every

[TABLE]

be partitioned into 4 blocks such that each $U_{i2}$ has the same size as $X_{i}$ (compare with (9)). Define the $\mathbb{G}$-linear mappings $\pi_{k}:\mathcal{U}_{k}\to\mathcal{U}$ ($k=1,2$) as follows:

[TABLE]

Then

[TABLE]

Fact 1: $\dim_{\mathbb{G}}\mathcal{U}_{1}=\dim_{\mathbb{G}}\mathcal{U}_{2}.$

Indeed, for the $t$-tuple (10) from $\mathcal{U}_{1}$ and for every $(U_{1},\dots,U_{t})\in\mathcal{U}_{2}$, we have

[TABLE]

Hence $(U_{1},\dots,U_{t})\mapsto(U_{1}P_{1},\dots,U_{t}P_{t})$ is a $\mathbb{G}$-linear bijection $\mathcal{U}_{2}\to\mathcal{U}_{1}$, which proves Fact 1.

Fact 2: $\operatorname{Ker}\pi_{1}=\operatorname{Ker}\pi_{2}.$

A $t$-tuple $U\in\mathcal{U}$ belongs to $\operatorname{Ker}\pi_{1}$ if and only if

[TABLE]

if and only if

[TABLE]

if and only if $U\in\operatorname{Ker}\pi_{2}$.

Fact 3: $\operatorname{Im}\pi_{1}\subset\operatorname{Im}\pi_{2}.$

For each

[TABLE]

there exist $U_{12},U_{14},\dots,U_{t2},U_{t4}$ such that

[TABLE]

which means that

[TABLE]

Then

[TABLE]

and so $U\in\operatorname{Im}\pi_{2},$ which proves Fact 3.

By (11) and Facts 1–3, $\operatorname{Im}\pi_{1}=\operatorname{Im}\pi_{2}$. Since $(I,\dots,I)\in\mathcal{U}_{2}$, $\left(\left[\begin{smallmatrix}I&0\\ 0&0\end{smallmatrix}\right],\dots,\left[\begin{smallmatrix}I&0\\ 0&0\end{smallmatrix}\right]\right)\in\operatorname{Im}\pi_{2}=\operatorname{Im}\pi_{1}$. Hence there are $U_{12},U_{14},\dots,U_{t2},U_{t4}$ such that

[TABLE]

which means that

[TABLE]

Equating the $(1,2)$ blocks in (12), we get $A_{i}U_{i^{\prime}2}^{\alpha_{i}}=C_{i}+U_{i^{\prime\prime}2}^{\beta_{i}}B_{i}.$ Thus, $(U_{12},\dots,U_{t2})$ is a solution of the system (7). ∎

Proof of Theorem 1.

$\Longrightarrow$. If $(\underline{X}_{1},\dots,\underline{X}_{t})$ is a solution of (2), then the equalities (3) hold for $P_{1},\dots,P_{t}$ defined in (9).

$\Longleftarrow$. Suppose there are nonsingular matrices $P_{1},\dots,P_{t}$ satisfying (3). We consider the set $\{1,\complement,\dagger,\Asterisk\}$ as the abelian group with multiplication

[TABLE]

that corresponds to the compositions of the matrix mappings $A\mapsto A^{\varepsilon}$, $\varepsilon\in\{1,\complement,\dagger,\Asterisk\}$.

Represent (3) in the form

[TABLE]

in which $\alpha_{i},\beta_{i}\in\{1,\complement\}$ and $\lambda_{i},\mu_{i}\in\{1,\dagger\}$ are such that $\alpha_{i}\lambda_{i}=\varepsilon_{i}$ and $\beta_{i}\mu_{i}=\delta_{i}$. Applying $\dagger$ to (13) and multiplying each factor by $J=\left[\begin{smallmatrix}0&I\\ -I&0\end{smallmatrix}\right]$ on the left and by $J^{-1}=\left[\begin{smallmatrix}0&-I\\ I&0\end{smallmatrix}\right]$ on the right, we get

[TABLE]

Using

[TABLE]

and (4), we rewrite (14) as follows:

[TABLE]

The equalities (13) and (15) and Lemma 1 ensure the solvability of the system formed by $2s$ matrix equations

[TABLE]

($i=1,\dots,s$) with unknown matrices $Y_{1,1},\dots,Y_{1,t},Y_{\dagger,1},\dots,Y_{\dagger,t}$. Let $\underline{Y}_{1,1},\dots,\underline{Y}_{1,t},$ $\underline{Y}_{\dagger,1},\dots,\underline{Y}_{\dagger,t}$ be its solution. Substituting these matrices to (16) and applying $\dagger$ to the right equalities, we get

[TABLE]

Adding the left and right equalities, we obtain

[TABLE]

Write $\underline{X}_{i}:=(\underline{Y}_{1,i}+\underline{Y}_{\dagger,i}^{\dagger})/2$ for $i=1,\dots,t$. Then $\underline{X}_{i}^{\dagger}=(\underline{Y}_{\dagger,i}+\underline{Y}_{1,i}^{\dagger})/2$. By (17),

[TABLE]

Therefore, $\underline{X}_{1},\dots,\underline{X}_{t}$ is a solution of the system (2). ∎

Acknowledgements

A. Dmytryshyn was supported by the Swedish Research Council (VR) grant E0485301, and by eSSENCE, a strategic collaborative e-Science programme funded by the Swedish Research Council. V. Futorny was supported by CNPq grant 301320/2013-6 and FAPESP grant 2014/09310-5. V.V. Sergeichuk was supported by FAPESP grant 2015/05864-9.