Throughout this paper, let $\mathbb{C}^{n\times m}$ be the set of all $n\times m$ complex matrices, $\mathbb{H}^{n\times n}$ denote the class of $n\times n$ Hermitian matrices, $\mathbb{U}^{n\times n}$ be the set of all $n\times n$ unitary matrices. Denote by $I_{n}$ the identity matrix with order $n$. Let $J=(e_{n}, e_{n-1},\cdots, e_{1})$, where $e_{i}$ is the $i$th column of $I_{n}$. For matrix $A$, $A^{\ast}$, $A^{+}$, $\|A\|_{F}$ and $r(A)$ represent its conjugate transpose, Moore-Penrose inverse, Frobenius norm and rank, respectively. For a matrix $A$, the two matrices $L_{A}$ and $R_{A}$ stand for the two orthogonal projectors $L_{A}=I-A^{+}A$, $R_{A}=I-AA^{+}$ induced by $A$.

Definition 1 Let $R\in \mathbb{C}^{n\times n}$ be a nontrivial unitary involution, i.e., $R=R^{\ast}=R^{-1}\neq I_{n}$. We say that $A\in \mathbb{C}^{n\times n}$ is a Hermitian $R$-symmetric matrix, if $A^{\ast}=A, RAR=A$. We denoted by $\mathbb{HRS}^{n\times n}$ the set of all $n\times n$ Hermitian $R$-symmetric matrices.

In matrix theory and applications, many problems are closely related to the ranks of some matrix expressions with variable entries, and so it is necessary to explicitly characterize the possible ranks of the matrix expressions concerned. The study on the possible ranks of matrix equations can be traced back to the late 1970s (see, e.g. [1-3]). Recently, the extremal ranks, i.e., maximal and minimal ranks, of some matrix expressions have found many applications in control theory [4, 5], statistics, and economics (see, e.g. [6, 7]).

In this paper, we consider the Hermitian $R$-symmetric extremal rank solutions of the matrix equation

$\begin{eqnarray}\label{firstfor} AX=B, \end{eqnarray}$

(1.1)

where $X$ and $B$ are given matrices in $\mathbb{C}^{n\times m}$.

We also consider the matrix nearness problem

$\begin{eqnarray} \min\limits_{A\in S_{m}}\left\|A-\tilde{A}\right\|_{F}, \end{eqnarray}$

(1.2)

where $\tilde{A}$ is a given matrix in $\mathbb{C}^{n\times m}$ and $S_{m}$ is the minimal rank solution set of eq. (1.1).

We first know that Hermitian $R$-symmetric matrices have the following properties.

Since $R=R^{\ast}=R^{-1}\neq I_{n}$, the only possible eigenvalues of $R$ are $+1$ and $-1$. Let $r$ and $s$ be respectively the dimensions of the eigenspaces of $R$ associated with the eigenvalues $\lambda=1$, and $\lambda=-1$; thus $r,s>1$ and $r+s=n$. Let

$\begin{equation} P=\left[p_{1} \ \ \ \cdots \ \ \ p_{r}\right] \ \ \ {\rm and} \ \ \ Q=\left[q_{1} \ \ \ \cdots \ \ \ q_{s}\right], \end{equation}$

(2.1)

where $\{p_{1}, \ldots, p_{r}\}$ and $\{q_{1}, \ldots, q_{s}\}$ are orthonormal bases for the eigenspaces. $P$ and $Q$ can be found by applying the Gram-Schmidt process to the columns of $I+R$ and $I-R$, respectively.

Lemma 1 [8] $A\in \mathbb{C}^{n\times n}$ is Hermitian and $R$-symmetric if and only if

$\begin{eqnarray} A=\left[\begin{array}{cc}P \ & \ Q\end{array}\right]\left[\begin{array}{cc}A_{P}&0\\0&A_{Q}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right] \end{eqnarray}$

(2.2)

with $A_{P}=P^{\ast}AP\in \mathbb{H}^{r\times r}$, $A_{Q}=Q^{\ast}AQ\in \mathbb{H}^{s\times s}$.

Given matrix $X_{1}, B_{1}\in \mathbb{C}^{n\times m}$, the singular value decomposition of $X_{1}$ be

$\begin{equation} X_{1}=U_{1}\left[\begin{array}{cc}\Sigma_{1} & 0\\0 & 0\end{array}\right]V_{1}^{\ast}=U_{11}\Sigma_{1} V_{11}^{\ast}, \end{equation}$

(2.3)

where $U_{1}=[U_{11},U_{12}]\in \mathbb{U}^{n\times n}$, $U_{11}\in \mathbb{C}^{n\times r_{1}}$, $V_{1}=[V_{11},V_{12}]\in \mathbb{U}^{m\times m}$, $V_{11}\in \mathbb{C}^{m\times r_{1}}$, $r_{1}=r(X_{1})$, $\Sigma_{1}={\rm diag}(\sigma_{1},\ldots,\sigma_{r_{1}})$, $\sigma_{1}\geq\cdots\geq\sigma_{r_{1}}>0$.

Let $A_{11}=U_{11}^{\ast}B_{1}V_{11}\Sigma_{1}^{-1}$, $A_{12}=U_{12}^{\ast}B_{1}V_{11}\Sigma_{1}^{-1}$, $G_{1}=A_{12}L_{A_{11}}$, the singular value decomposition of $G_{1}$ be

$\begin{eqnarray} G_{1}=P_{1}\left[\begin{array}{cc}\Gamma_{1} & 0\\0 & 0\end{array}\right]Q_{1}^{\ast}=P_{11}\Gamma_{1} Q_{11}^{\ast}, \end{eqnarray}$

(2.4)

where $P_{1}=[P_{11},P_{12}]\in \mathbb{U}^{(n-r_{1})\times (n-r_{1})}$, $P_{11}\in \mathbb{C}^{(n-r_{1})\times s_{1}}$, $Q_{1}=[Q_{11},Q_{12}]\in \mathbb{U}^{r_{1}\times r_{1}}$, $Q_{11}\in \mathbb{C}^{r_{1}\times s_{1}}$, $s_{1}=r(G_{1})$, $\Gamma_{1}={\rm diag}(\gamma_{1},\ldots,\gamma_{s_{1}})$, $\gamma_{1}\geq\cdots\geq\gamma_{s_{1}}>0$.

Lemma 2 [9] Given matrices $X_{1},B_{1}\in \mathbb{C}^{n\times m}$. Let the singular value decompositions of $X_{1}$ and $G_{1}$ be $(2.3)$, $(2.4)$, respectively. Then the matrix equation $A_{1}X_{1}=B_{1}$ has a solution $A_{1}$ in $\mathbb{H}^{n\times n}$ if and only if

$\begin{eqnarray} X_{1}^{\ast}B_{1}=B_{1}^{\ast}X_{1},\ \ \ \ B_{1}X_{1}^{+}X_{1}=B_{1}. \end{eqnarray}$

(2.5)

In this case, let $\Omega_{1}$ be the set of all Hermitian solutions of equation $A_{1}X_{1}=B_{1}$, then the extreme ranks of $A_{1}$ are as follows:

(1) The maximal rank of $A_{1}$ is

$\begin{eqnarray} \max\limits_{A_{1}\in \Omega_{1}}r(A_{1})=n+r(B_{1})-r(X_{1}). \end{eqnarray}$

(2.6)

The general expression of $A_{1}$ satisfying $(2.6)$ is

$\begin{eqnarray} A_{1}=A_{0}+U_{12}N_{1}U^{\ast}_{12} \end{eqnarray}$

(2.7)

where $A_{0}=B_{1}X^{+}_{1}+(B_{1}X^{+}_{1})^{+}R_{X_{1}}+R_{X_{1}}B_{1}X^{+}_{1}(X_{1}X^{+}_{1}B_{1}X^{+}_{1})^{+}(B_{1}X^{+}_{1})^{\ast}R_{X_{1}}$ and $N_{1}\in \mathbb{H}^{(n-r_{1})\times (n-r_{1})}$ is chosen such that $r(R_{G_{1}}N_{1}R_{G_{1}})=n+r(X^{\ast}_{1}B_{1})-r(B_{1})-r(X_{1})$.

(2) The minimal rank of $A_{1}$ is

$\begin{eqnarray} \min\limits_{A_{1}\in \Omega_{1}}r(A_{1})=2r(B_{1})-r(X^{\ast}_{1}B_{1}). \end{eqnarray}$

(2.8)

The general expression of $A_{1}$ satisfying $(2.8)$ is

$\begin{eqnarray} A_{1}=A_{0}+U_{12}P_{11}P^{\ast}_{11}M_{1}P_{11}P^{\ast}_{11}U^{\ast}_{12} \end{eqnarray}$

(2.9)

where $A_{0}=B_{1}X^{+}_{1}+(B_{1}X^{+}_{1})^{+}R_{X_{1}}+R_{X_{1}}B_{1}X^{+}_{1}(X_{1}X^{+}_{1}B_{1}X^{+}_{1})^{+}(B_{1}X^{+}_{1})^{\ast}R_{X_{1}}$ and $M_{1}\in \mathbb{H}^{(n-r_{1})\times (n-r_{1})}$ is arbitrary.

Assume $P$, $Q$ with the forms of $(2.1)$. Let

$\begin{eqnarray} \left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right]X=\left[\begin{array}{ccc}X_{2} \\ X_{3}\end{array}\right],\ \ \left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right]B=\left[\begin{array}{ccc}B_{2} \\ B_{3}\end{array}\right], \end{eqnarray}$

(3.1)

where $X_{2}\in \mathbb{C}^{r\times m}$, $X_{3}\in \mathbb{C}^{s\times m}$, $B_{2}\in \mathbb{C}^{r\times m}$, $B_{3}\in \mathbb{C}^{s\times m}$, and the singular value decomposition of matrices $X_{2}$, $X_{3}$ are, respectively,

$\begin{equation} X_{2}=U_{2}\left[\begin{array}{cc}\Sigma_{2} & 0\\0 & 0\end{array}\right]V_{2}^{\ast}=U_{21}\Sigma_{2} V_{21}^{\ast}, \end{equation}$

(3.2)

where $U_{2}=[U_{21},U_{22}]\in \mathbb{U}^{r\times r}$, $U_{21}\in \mathbb{C}^{r\times r_{2}}$, $V_{2}=[V_{21},V_{22}]\in \mathbb{U}^{m\times m}$, $V_{21}\in \mathbb{C}^{m\times r_{2}}$, $r_{2}=r(X_{2})$, $\Sigma_{2}={\rm diag}(\alpha_{1},\ldots,\alpha_{r_{2}})$, $\alpha_{1}\geq\cdots\geq\alpha_{r_{2}}>0$,

$\begin{equation} X_{3}=U_{3}\left[\begin{array}{cc}\Sigma_{3} & 0\\0 & 0\end{array}\right]V_{3}^{T}=U_{31}\Sigma_{3} V_{31}^{T}, \end{equation}$

(3.3)

where $U_{3}=[U_{31},U_{32}]\in \mathbb{U}^{s\times s}$, $U_{31}\in \mathbb{C}^{s\times r_{3}}$, $V_{3}=[V_{31},V_{32}]\in \mathbb{U}^{m\times m}$, $V_{31}\in \mathbb{C}^{m\times r_{3}}$, $r_{3}=r(X_{3})$, $\Sigma_{3}={\rm diag}(\beta_{1},\cdots,\beta_{r_{3}})$, $\beta_{1}\geq\cdots\geq\beta_{r_{3}}>0$.

Let $A_{21}=U_{21}^{\ast}B_{2}V_{21}\Sigma_{2}^{-1}$, $A_{22}=U_{22}^{\ast}B_{2}V_{21}\Sigma_{2}^{-1}$, $G_{2}=A_{22}L_{A_{21}}$, $A_{31}=U_{31}^{\ast}B_{3}V_{31}\Sigma_{3}^{-1}$, $A_{32}=U_{32}^{\ast}B_{3}V_{31}\Sigma_{3}^{-1}$, $G_{3}=A_{32}L_{A_{31}}$, the singular value decomposition of matrices $G_{2}$, $G_{3}$ are, respectively,

$\begin{eqnarray} G_{2}=P_{2}\left[\begin{array}{cc}\Gamma_{2} & 0\\0 & 0\end{array}\right]Q_{2}^{\ast}=P_{21}\Gamma_{2} Q_{21}^{\ast}, \end{eqnarray}$

(3.4)

where $P_{2}=[P_{21},P_{22}]\in \mathbb{U}^{(r-r_{2})\times (r-r_{2})}$, $P_{21}\in \mathbb{C}^{(r-r_{2})\times s_{2}}$, $Q_{2}=[Q_{21},Q_{22}]\in \mathbb{U}^{r_{2}\times r_{2}}$, $Q_{21}\in \mathbb{C}^{r_{2}\times s_{2}}$, $s_{2}=r(G_{2})$, $\Gamma_{2}={\rm diag}(\zeta_{1},\ldots,\zeta_{s_{2}})$, $\zeta_{1}\geq\cdots\geq\zeta_{s_{2}}>0$.

$\begin{eqnarray} G_{3}=P_{3}\left[\begin{array}{cc}\Gamma_{3} & 0\\0 & 0\end{array}\right]Q_{3}^{\ast}=P_{31}\Gamma_{3} Q_{31}^{\ast}, \end{eqnarray}$

(3.5)

where $P_{3}=[P_{31},P_{32}]\in \mathbb{U}^{(s-r_{3})\times (s-r_{3})}$, $P_{31}\in \mathbb{C}^{(s-r_{3})\times s_{3}}$, $Q_{3}=[Q_{31},Q_{32}]\in \mathbb{U}^{r_{3}\times r_{3}}$, $Q_{31}\in \mathbb{C}^{r_{3}\times s_{3}}$, $s_{3}=r(G_{3})$, $\Gamma_{3}={\rm diag}(\xi_{1},\cdots,\xi_{s_{3}})$, $\xi_{1}\geq\cdots\geq\xi_{s_{3}}>0$.

Now we can establish the existence theorems as follows.

Theorem 1 Let $X,B\in \mathbb{C}^{n\times m}$ be known. Suppose $P$, $Q$ with the forms of $(2.1)$, $\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right]X$, $\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right]B$ have the partition forms of $(3.1)$, and the singular value decompositions of the matrices $X_{2}$, $X_{3}$ and $G_{2}$, $G_{3}$ are given by $(3.2)$, $(3.3)$ and $(3.4)$, $(3.5)$, respectively. Then equation $(1.1)$ has a solution $A\in \mathbb{HRS}^{n\times n}$ if and only if

$\begin{eqnarray} X_{2}^{\ast}B_{2}=B_{2}^{\ast}X_{2},\ \ B_{2}X_{2}^{+}X_{2}=B_{2},\ \ X_{3}^{\ast}B_{3}=B_{3}^{\ast}X_{3},\ \ B_{3}X_{3}^{+}X_{3}=B_{3}. \end{eqnarray}$

(3.6)

In this case, let $\Omega$ be the set of all Hermitian $R$-symmetric solutions of equation $(1.1)$, then the extreme ranks of $A$ are as follows:

(1) The maximal rank of $A$ is

$\begin{eqnarray} \max\limits_{A \in \Omega}r(A)=n+r(B_{2})+r(B_{3})-r(X_{2})-r(X_{3}). \end{eqnarray}$

(3.7)

The general expression of $A$ satisfying $(3.7)$ is

$\begin{eqnarray} A=\left[\begin{array}{cc}P \ & \ Q \end{array}\right]\left[\begin{array}{cc}A_{2}+U_{22}N_{2}U^{\ast}_{22} & 0\\0 & A_{3}+U_{32}N_{3}U^{\ast}_{32}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right], \end{eqnarray}$

(3.8)

where

$A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{\ast}R_{X_{i}},$

$ i=2,3$ and $N_{2}\in \mathbb{H}^{(r-r_{2})\times (r-r_{2})}$, $N_{3}\in \mathbb{H}^{(s-r_{3})\times (s-r_{3})}$ are chosen such that

$\begin{eqnarray*}&& r(R_{G_{2}}N_{2}R_{G_{2}})=r+r(X^{\ast}_{2}B_{2})-r(B_{2})-r(X_{2}),\\ && r(R_{G_{3}}N_{3}R_{G_{3}})=s+r(X^{\ast}_{3}B_{3})-r(B_{3})-r(X_{3}).\end{eqnarray*}$

(2) The minimal rank of $A$ is

$\begin{eqnarray} \min\limits_{A\in \Omega}r(A)=2r(B_{2})+2r(B_{3})-r(X^{\ast}_{2}B_{2})-r(X^{\ast}_{3}B_{3}). \end{eqnarray}$

(3.9)

The general expression of $A$ satisfying $(3.9)$ is

$\begin{eqnarray} A=\left[P \ \ Q \right]\left[\begin{array}{cc}A_{2}+U_{22}P_{21}P^{T}_{21}M_{2}P_{21}P^{T}_{21}U^{T}_{22} & 0\\0 & A_{3}+U_{32}P_{31}P^{T}_{31}M_{3}P_{31}P^{T}_{31}U^{T}_{32}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right],\nonumber\\ \end{eqnarray}$

(3.10)

where $A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{\ast}R_{X_{i}}, i=2,3$, and $M_{2}\in \mathbb{H}^{(r-r_{2})\times (r-r_{2})}$, $M_{3}\in \mathbb{H}^{(s-r_{3})\times (s-r_{3})}$ are arbitrary.

Proof  Suppose the matrix equation (1.1) has a solution $A$ which is Hermitian $R$-symmetric, then it follows from Lemma 1 that there exist $A_{P} \in \mathbb{H}^{r\times r}$, $A_{Q} \in \mathbb{H}^{s\times s}$ satisfying

$\begin{eqnarray} A=\left[\begin{array}{cc}P \ & \ Q\end{array}\right]\left[\begin{array}{cc}A_{P}&0\\0&A_{Q}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right] \ \ \ {\rm and }\ \ \ AX=B. \end{eqnarray}$

(3.11)

By (3.1), that is

$\begin{eqnarray} \left[\begin{array}{cc}A_{P}&0\\0&A_{Q}\end{array}\right]\left[\begin{array}{cc}X_{2}\\X_{3}\end{array}\right] =\left[\begin{array}{cc}B_{2}\\B_{3}\end{array}\right], \end{eqnarray}$

(3.12)

i.e.,

$\begin{eqnarray} A_{P}X_{2}=B_{2}, A_{Q}X_{3}=B_{3}. \end{eqnarray}$

(3.13)

Therefore by Lemma 2, (3.6) hold, and in this case, let $\Omega$ be the set of all Hermitian $R$-symmetric solutions of equation $(1.1)$, we have

(1) By (3.1),

$\begin{eqnarray} \max\limits_{A\in \Omega}r(A)=\max\limits_{ { {{\scriptscriptstyle A_{P}X_{2}=B_{2}\atop { {A_{P}^{\ast}=A_{2}}}}} }}r(A_{P})+ \max\limits_{ { {{\scriptscriptstyle A_{Q}X_{3}=B_{3}\atop { {A^{\ast}_{Q}=A_{3}}}}} }}r(A_{Q}). \end{eqnarray}$

(3.14)

By Lemma 2,

$\begin{eqnarray} \max\limits_{ { {{\scriptscriptstyle A_{P}X_{2}=B_{2}\atop { {A_{P}^{\ast}=A_{2}}}}} }}r(A_{P})=r+r(B_{2})-r(X_{2}) \end{eqnarray}$

(3.15)

and

$\begin{eqnarray} \max\limits_{ { {{\scriptscriptstyle A_{Q}X_{3}=B_{3}\atop { {A^{\ast}_{Q}=A_{3}}}}} }}r(A_{Q})=s+r(B_{3})-r(X_{3}). \end{eqnarray}$

(3.16)

Taking (3.15) and (3.16) into (3.14) yields (3.7).

According to the general expression of the solution in Lemma 2, it is easy to verify the rest of part in (1).

(2) The proof is very similar to that of (1) By (3.1) and Lemma 2, so we omit it.

From (3.10), when the solution set $S_{m}=\{A\mid AX=B, A\in \mathbb{HRS}^{n\times n}, r(A)=\min\limits_{Y\in \Omega}r(Y)\}$ is nonempty, it is easy to verify that $S_{m}$ is a closed convex set, therefore there exists a unique solution $\hat{A}$ to the matrix nearness problem (1.2).

Theorem 2 Given matrix $\tilde{A}$, and the other given notations and conditions are the same as in Theorem 1. Let

$\begin{eqnarray} \left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right]\tilde{A}\left[\begin{array}{cc}P \ & \ Q\end{array}\right]=\left[\begin{array}{cc}\tilde{A}_{11}&\tilde{A}_{12}\\\tilde{A}_{21}&\tilde{A}_{22}\end{array}\right], \tilde{A}_{11} \in \mathbb{C}^{r\times r}, \tilde{A}_{22} \in \mathbb{C}^{s\times s}, \end{eqnarray}$

(3.1)

and we denote

$\begin{eqnarray} U^{\ast}_{2}(\tilde{A}_{11}-A_{2})U_{2}=\left[\begin{array}{cc}\tilde{B}_{11} & \tilde{B}_{12}\\\tilde{B}_{21} & \tilde{B}_{22}\\ \end{array}\right], \tilde{B}_{11} \in \mathbb{C}^{r_{2}\times r_{2}}, \tilde{B}_{22} \in \mathbb{C}^{(r-r_{2})\times (r-r_{2})}, \end{eqnarray}$

(3.2)

$\begin{eqnarray} U^{\ast}_{3}(\tilde{A_{22}}-A_{3})U_{3}=\left[\begin{array}{ccc}\tilde{C}_{11} & \tilde{C}_{12}\\\tilde{C}_{21} & \tilde{C}_{22}\\ \end{array}\right], \tilde{C}_{11} \in \mathbb{C}^{r_{3}\times r_{3}}, \tilde{C}_{22} \in \mathbb{C}^{(s-r_{3})\times (s-r_{3})}. \end{eqnarray}$

(3.3)

If $S_{m}$ is nonempty, then problem $(1.2)$ has a unique $\hat{A}$ which can be represented as

$\begin{eqnarray} \hat{A}=[P\ \ Q]\left[\begin{array}{cc}A_{2}+U_{22}P_{21}P^{\ast}_{21}\tilde{B}_{22}P_{21}P^{\ast}_{21}U^{\ast}_{22} & 0\\0 & A_{3}+U_{32}P_{31}P^{\ast}_{31}\tilde{C}_{22}P_{31}P^{\ast}_{31}U^{\ast}_{32}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right], \end{eqnarray}$

(3.4)

where $\tilde{B}_{22}$, $\tilde{C}_{22}$ are the same as in $(3.2)$, $(3.3)$.

Proof When $S_{m}$ is nonempty, it is easy to verify from (3.10) that $S_{m}$ is a closed convex set. Problem (1.2) has a unique solution $\hat{A}$. By Theorem 1, for any $A\in S_{m}$, $A$ can be expressed as

$\begin{eqnarray} A=[P \ \ Q]\left[\begin{array}{cc}A_{2}+U_{22}P_{21}P^{\ast}_{21}M_{2}P_{21}P^{\ast}_{21}U^{\ast}_{22} & 0\\0 & A_{3}+U_{32}P_{31}P^{\ast}_{31}M_{3}P_{31}P^{\ast}_{31}U^{\ast}_{32}\end{array}\right]\left[\begin{array}{cc}P^{\ast} \\ Q^{\ast}\end{array}\right], \end{eqnarray}$

(3.5)

where

$A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{\ast}R_{X_{i}},$

$ i=2,3$, and $M_{2}\in \mathbb{H}^{(k-r_{2})\times (r-r_{2})}$, $M_{3}\in \mathbb{H}^{(s-r_{3})\times (s-r_{3})}$ are arbitrary.

Using the invariance of the Frobenius norm under unitary transformations, and

$P_{21}P_{21}^{\ast}+P_{22}P_{22}^{\ast}=I, P_{31}P_{31}^{\ast}+P_{32}P_{32}^{\ast}=I, $

where $P_{21}P_{21}^{\ast}$, $P_{22}P_{22}^{\ast}$, $P_{31}P_{31}^{\ast}$, $P_{32}P_{32}^{\ast}$ are unitary projection matrices, and

$P_{21}P_{21}^{\ast}P_{22}P_{22}^{\ast}=0, P_{31}P_{31}^{\ast}P_{32}P_{32}^{\ast}=0,$

we have

$\begin{aligned} \|\tilde{A}-A\|_{F}^2 =& \left\|\tilde{A}_{11}-A_{2}-U_{22}P_{21}P^{\ast}_{21}M_{2}P_{21}P^{\ast}_{21}U^{T}_{22}\right\|_{F}^2+ \left\|\tilde{A}_{12}\right\|_{F}^2\\ &+ \left\|\tilde{A}_{22}-A_{3}-U_{32}P_{31}P^{\ast}_{31}M_{3}P_{31}P^{\ast}_{31}U^{\ast}_{32}\right\|_{F}^2+ \left\|\tilde{A}_{21}\right\|_{F}^2\\ =& \left\|U^{\ast}_{2}(\tilde{A}_{11}-A_{2})U_{2}-\left[\begin{array}{cc}0 & 0\\0 & P_{21}P^{\ast}_{21}M_{2}P_{21}P^{\ast}_{21}\end{array}\right]\right\|_{F}^2+ \left\|\tilde{A}_{12}\right\|_{F}^2\\ &+ \left\|U^{\ast}_{3}(\tilde{A}_{22}-A_{3})U_{3}-\left[\begin{array}{cc}0 & 0\\0 & P_{31}P^{\ast}_{31}M_{3}P_{31}P^{\ast}_{31}\end{array}\right]\right\|_{F}^2+ \left\|\tilde{A}_{21}\right\|_{F}^2\\ =& \left\|\tilde{A}_{12}\right\|_{F}^2 +\left\|\tilde{A}_{21}\right\|_{F}^2+\left\|\tilde{B}_{11}\right\|_{F}^2+\left\|\tilde{B}_{12}\right\|_{F}^2+\left\|\tilde{B}_{21}\right\|_{F}^2 +\left\|\tilde{C}_{11}\right\|_{F}^2\\ &+ \left\|\tilde{C}_{12}\right\|_{F}^2+\left\|\tilde{B}_{22}P_{22}P^{\ast}_{22}\right\|_{F}^2 +\left\|P_{22}P^{\ast}_{22}\tilde{B}_{22}P_{21}P^{\ast}_{21}\right\|_{F}^2\\ &+ \left\|P_{21}P^{\ast}_{21}\tilde{B}_{22}P_{21}P^{\ast}_{21}-P_{21}P^{\ast}_{21}M_{2}P_{21}P^{\ast}_{21}\right\|_{F}^2\\ &+ \left\|\tilde{C}_{21}\right\|_{F}^2+\left\|\tilde{C}_{22}P_{32}P^{\ast}_{32}\right\|_{F}^2 +\left\|P_{32}P^{\ast}_{32}\tilde{C}_{22}P_{31}P^{\ast}_{31}\right\|_{F}^2\\ &+ \left\|P_{31}P^{\ast}_{31}\tilde{C}_{22}P_{31}P^{\ast}_{31}-P_{31}P^{\ast}_{31}M_{3}P_{31}P^{\ast}_{31}\right\|_{F}^2. \end{aligned}$

Therefore, $\min\limits_{A\in S_{m}}\|\tilde{A}-A\|_{F}$ is equivalent to

$\begin{eqnarray} && \min\limits_{M_{2}\in \mathbb{H}^{(r-r_{2})\times (r-r_{2})}}\left\|P_{21}P^{T}_{21}\tilde{B}_{22}P_{21}P^{\ast}_{21}-P_{21}P^{\ast}_{21}M_{2}P_{21}P^{\ast}_{21}\right\|_{F}, \end{eqnarray}$

(3.6)

$\begin{eqnarray} && \min\limits_{M_{3}\in \mathbb{H}^{(s-r_{3})\times (s-r_{3})}}\left\|P_{31}P^{\ast}_{31}\tilde{C}_{22}P_{31}P^{\ast}_{31}-P_{31}P^{\ast}_{31}M_{3}P_{31}P^{\ast}_{31}\right\|_{F}. \end{eqnarray}$

(3.7)

Obviously, the solutions of (3.6), (3.7) can be written as

$\begin{eqnarray} && M_{2}=\tilde{B}_{22}+P_{22}P^{\ast}_{22}\tilde{M}_{2}P_{22}P^{\ast}_{22},\ \ \forall \tilde{M_{2}}\in \mathbb{H}^{(r-r_{2})\times (r-r_{2})}, \end{eqnarray}$

(3.8)

$\begin{eqnarray} && M_{3}=\tilde{C}_{22}+P_{32}P^{\ast}_{32}\tilde{M}_{3}P_{32}P^{\ast}_{32},\ \ \forall \tilde{M_{3}}\in \mathbb{H}^{(s-r_{3})\times (s-r_{3})}. \end{eqnarray}$

(4.9)

Substituting (3.8), (3.9) into (3.5), then we get that the unique solution to problem (1.2) can be expressed in (3.4). The proof is completed.