Throughout this paper, let $R^{n\times m}$ be the set of all n £ m real matrices, $SR^{n\times m}$ be the set of all ${n\times m}$ real symmetric matrices, $OR^{n\times n}$ be the set of all ${n\times n}$ orthogonal matrices.Denote by I_n the identity matrix with order n. For matrix $A,{\rm{ }}{A^T},{\rm{ }}{A^ + },{\rm{ }}\left\| A \right\|$ and r(A)represent its 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 ${L_A}{\rm{ }} = {\rm{ }}I - {A^ + }A,{R_A}{\rm{ }} = {\rm{ }}I{\rm{ }} - {\rm{ }}A{A^ + }$ induced by A.

Definition 1 A real symmetric matrix $A=(a_{ij})\in R^{n\times n}$ is said to be a Bisymmetric matrix if $a_{ij}=a_{n+1-j,n+1-i}, i,j=1,2,\cdots,n$. The set of all $n\times n$ Bisymmetric matrices is denoted by $BSR^{n\times n}$.

In this paper, we consider the Bisymmetric extremal rank solutions of the matrix equation

${\rm{ }}AX = B$

(1.1)

where X and B are given matrices in $R^{n\times m}$. In 1972, Mitra [1] considered solutions with fixed ranks for the matrix equations $AX=B$ and $AXB=C$. In 1984, Mitra [2] gave common solutions of minimal rank of the pair of complex matrix equations $AX=C, XB=D$. In 1987, Uhlig [3] presented the extremal ranks of solutions to the matrix equation $AX=B$. In 1990, Mitra studied the minimal ranks of common solutions to the pair of matrix equations $A_{1}X_{1}B_{1}=C_{1}$ and $A_{2}X_{2}B_{2}=C_{2}$ over a general field in [4]. In 2003, Tian (see [5, 6]) investigated the extremal rank solutions to the complex matrix equation $AXB=C$ and gave some applications. Xiao et al. [7, 8] considered the symmetric and anti-symmetric minimal rank solution to equation $AX=B$. The Bisymmetric maximal and minimal rank solutions of the matrix equation (1.1), however, has not been considered yet. In this paper, we will discuss this problem.

We also consider the matrix nearness problem

$\min\limits_{A\in S_{m}}\left\|A-\tilde{A}\right\|,$

(1.2)

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

Denote by $e_{i}$ be the $i$th column of $I_{n}$ and set $S_{n}=(e_{n},e_{n-1},\cdots,e_{1})$. It is easy to see that

$S_{n}^{T}=S_{n},\ \ S_{n}^{T}S_{n}=I.$

Let $k=[\frac{n}{2}]$, where $[\frac{n}{2}]$ is the maximum integer which is not greater than $\frac{n}{2}$. Define $D_{n}$ as

${D_n} = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} {{I_k}}&{{I_k}} \\ {{S_k}}&{ - {S_k}} \end{array}} \right](n = 2k),{D_n} = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} {{{\rm I}_k}}&0&{{I_k}} \\ 0&{\sqrt 2 }&0 \\ {{S_k}}&0&{ - {S_k}} \end{array}} \right](n = 2k + 1),$

(2.1)

then it is easy verified that the above matrices $D_{n}$ are orthogonal matrices.

Lemma 1 [9]Let $A\in R^{n\times n}$ and $D_{n}$ with the forms of $(2.1)$, then $A$ is the Bisymmetric matrix if and only if there exist $A_{2}\in SR^{(n-k)\times (n-k)}$ and $A_{3}\in SR^{k\times k}$, whether n is odd or even, such that

$A=D_{n}\left[\begin{array}{cc}A_{2}&0\\0&A_{3}\end{array}\right]D^{T}_{n}.$

(2.2)

Here, we always assume $k=[\frac{n}{2}]$.

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

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

(2.3)

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

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

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

(2.4)

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

Lemma 2 [7]Given matrices $X_{1},B_{1}\in R^{n\times m}$. 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 symmetric solution $A_{1}$ if and only if

$X_{1}^{T}B_{1}=B_{1}^{T}X_{1}, B_{1}X_{1}^{+}X_{1}=B_{1}.$

(2.5)

In this case, let $\Omega_{1}$ be the set of all symmetric 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

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

(2.6)

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

$A_{1}=A_{0}+U_{12}N_{1}U^{T}_{12},$

(2.7)

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

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

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

(2.8)

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

$A_{1}=A_{0}+U_{12}P_{11}P^{T}_{11}M_{1}P_{11}P^{T}_{11}U^{T}_{12},$

(2.9)

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

Assume D_n with the form of (2.1). Let

$D_n^TX = \left[ {_{{X_3}}^{{X_2}}} \right],D_n^TB = \left[ {_{{B_3}}^{{B_2}}} \right],$

(3.1)

where ${X_2} \in {R^{(n - k) \times m}},{X_3} \in {R^{k \times m}},{B_2} \in {R^{(n - k) \times m}},{B_3} \in {R^{k \times m}}$, and the singular value decomposition of matrices X₂, X₃ are, respectively,

${X_2} = {U_2}\left[ {\begin{array}{*{20}{c}} {{{\sum {} }_2}}&0\\ 0&0 \end{array}} \right]V_2^T = {U_{21}}\sum {_2} V_{21}^T,$

(3.2)

where ${U_2} = \left[ {{U_{21}},{U_{22}}} \right] \in O{R^{(n - k) \times (n - k)}},{U_{21}} \in {R^{(n - k) \times r2}},{V_2} = \left[ {{V_{21}},{V_{22}}} \right] \in O{R^{m \times m}},{V_{21}} \in O{R^{m \times r2}},r2 = r({X_2}),$${\sum {} _3} = {\rm{diag}}({\alpha _1}, \cdots ,{\alpha _{r2}}),{\alpha _1} \ge \cdots \ge {\alpha _{r2}} > 0,$.

${X_3} = {U_3}\left[ {\begin{array}{*{20}{c}} {{{\sum {} }_3}}&0\\ 0&0 \end{array}} \right]V_3^T = {U_{31}}{\sum {} _3}V_{31}^T,$

(3.3)

where ${U_3} = \left[ {{U_{31}},{U_{32}}} \right] \in O{R^{k \times k}},{U_{31}} \in {R^{(k \times r3}},{V_3} = \left[ {{V_{31}},{V_{32}}} \right] \in O{R^{m \times m}},$${V_{31}} \in O{R^{m \times r3}},r3 = r({X_3}),{\sum {} _3} = diang({\alpha _1}, \cdots ,{\alpha _{r2}}),{\alpha _1} \ge \cdots \ge {\alpha _{r2}} > 0,$.

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

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

(3.4)

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

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

(3.5)

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

Now we can establish the existence theorems as follows.

Theorem 1 Let $X,B\in R^{n\times m}$ be known. Suppose $D_{n}$ with the form of $(2.1)$, $D^{T}_{n}X$, $D^{T}_{n}B$ 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 the equation $(1.1)$ has a Bisymmetric solution $A$ if and only if

$X_{2}^{T}B_{2}=B_{2}^{T}X_{2},\ \ B_{2}X_{2}^{+}X_{2}=B_{2},\ \ X_{3}^{T}B_{3}=B_{3}^{T}X_{3},\ \ B_{3}X_{3}^{+}X_{3}=B_{3}.$

(3.6)

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

$(1)$ The maximal rank of $A$ is

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

(3.7)

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

$A=D_{n}\left[\begin{array}{cc}A_{2}+U_{22}N_{2}U^{T}_{22} & 0\\0 & A_{3}+U_{32}N_{3}U^{T}_{32}\end{array}\right]D^{T}_{n}$

(3.8)

where $A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{T}R_{X_{i}}, \ \ i=2,3$, and $N_{2}\in SR^{(n-k-r_{2})\times (n-k-r_{2})}$, $N_{3}\in SR^{(k-r_{3})\times (k-r_{3})}$ are chosen such that

$r(R_{G_{2}}N_{2}R_{G_{2}})=n-k+r(X^{T}_{2}B_{2})-r(B_{2})-r(X_{2}),\\ r(R_{G_{3}}N_{3}R_{G_{3}})=k+r(X^{T}_{3}B_{3})-r(B_{3})-r(X_{3}).$

$(2)$ The minimal rank of $A$ is

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

(3.9)

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

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

(3.10)

where $A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{T}R_{X_{i}}, i=2,3$ and $M_{2}\in SR^{(n-k-r_{2})\times (n-k-r_{2})}$, $M_{3}\in SR^{(k-r_{3})\times (k-r_{3})}$ are arbitrary.

Proof Suppose the matrix equation (1.1) has a solution $A$ which is Bisymmetric, then it follows from Lemma 1 that there exist $A_{2} \in SR^{(n-k)\times (n-k)}$, $A_{3} \in SR^{k\times k}$ satisfying

$A=D_{n}\left[\begin{array}{cc}A_{2}&0\\0&A_{3}\end{array}\right]D_{n}^{T} \ \ \ {\rm and} \ \ \ AX=B. $

(3.11)

By (3.1), that is

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

(3.12)

i.e.,

$A_{2}X_{2}=B_{2},\ \ A_{3}X_{3}=B_{3}.$

(3.13)

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

(1) By (3.11),

$_{A \in \Omega }^{\max r}(A) = \mathop {\max r}\limits_{{\rm{ }}\begin{array}{*{20}{c}} {{A_2}{X_2} = {B_2}}\\ {A_2^T = {A_2}} \end{array}} ({A_2}) + \mathop {\max r}\limits_{{\rm{ }}\begin{array}{*{20}{c}} {{A_3}{X_3} = {B_3}}\\ {A_3^T = {A_3}} \end{array}} ({A_3}).$

(3.14)

By Lemma 2,

$\mathop {\max r}\limits_{{\rm{ }}\begin{array}{*{20}{c}} {{A_2}{X_2} = {B_2}}\\ {A_2^T = {A_2}} \end{array}} ({A_2}) = n - k + r({B_2}) - r({X_2}),\mathop {\max r}\limits_{\begin{array}{*{20}{c}} {{A_3}{X_3} = {B_3}}\\ {A_3^T = {A_3}} \end{array}} ({A_3}) = k + r({B_3}) - r({X_3}).$

(3.15)

Taking (3.15) 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 1, so we omit it.

From (3.10), when the solution set $S_{m}=\{A\mid AX=B, A\in BSR^{n\times n}, r(A)=\min\limits_{Y\in \Omega}r(Y)\}$ 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

$D_{n}^{T}\tilde{A}D_{n}=\left[\begin{array}{cc}\tilde{A}_{11}&\tilde{A}_{12}\\\tilde{A}_{21}&\tilde{A}_{22}\end{array}\right], \ \ \tilde{A}_{11} \in R^{(n-k)\times (n-k)},\ \ \tilde{A}_{22} \in R^{k\times k},$

(4.1)

and we denote

$U^{T}_{2}(\tilde{A}_{11}-A_{2})U_{2}=\left[ \begin{array}{cc}\tilde{B}_{11} & \tilde{B}_{12}\\\tilde{B}_{21} & \tilde{B}_{22}\\ \end{array}\right], \ \ \tilde{B}_{11} \in R^{r_{2}\times r_{2}},\ \ \tilde{B}_{22} \in R^{(n-k-r_{2})\times (n-k-r_{2})},$

(4.2)

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

(4.3)

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

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

(4.4)

where $\tilde{B}_{22}$, $\tilde{C}_{22}$ are the same as in (4.2), (4.3).

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 [10]. By Theorem 1, for any $A\in S_{m}$, $A$ can be expressed as

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

(4.5)

where $A_{i}=B_{i}X^{+}_{i}+(B_{i}X^{+}_{i})^{+}R_{X_{i}}+R_{X_{i}}B_{i}X^{+}_{i}(X_{i}X^{+}_{i}B_{i}X^{+}_{i})^{+}(B_{i}X^{+}_{i})^{T}R_{X_{i}}, \ \ i=2,3$, and $M_{2}\in SR^{(n-k-r_{2})\times (n-k-r_{2})}$, $M_{3}\in SR^{(k-r_{3})\times (k-r_{3})}$ are arbitrary.

Using the invariance of the Frobenius norm under orthogonal transformations, and $P_{21}P_{21}^{T}+P_{22}P_{22}^{T}=I$, $P_{31}P_{31}^{T}+P_{32}P_{32}^{T}=I$, where $P_{21}P_{21}^{T}$, $P_{22}P_{22}^{T}$, $P_{31}P_{31}^{T}$, $P_{32}P_{32}^{T}$ are orthogonal projection matrices, and $P_{21}P_{21}^{T}P_{22}P_{22}^{T}=0$, $P_{31}P_{31}^{T}P_{32}P_{32}^{T}=0$, we have

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

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

$\min\limits_{M_{2}\in SR^{(n-k)\times (n-k)}}\left\|P_{21}P^{T}_{21}\tilde{B}_{22}P_{21}P^{T}_{21}-P_{21}P^{T}_{21}M_{2}P_{21}P^{T}_{21}\right\|,$

(4.6)

$\min\limits_{M_{3}\in SR^{k\times k}}\left\|P_{31}P^{T}_{31}\tilde{C}_{22}P_{31}P^{T}_{31}-P_{31}P^{T}_{31}M_{3}P_{31}P^{T}_{31}\right\|.$

(4.7)

Obviously, the solutions of (4.6), (4.7) can be written as

$M_{2}=\tilde{B}_{22}+P_{22}P^{T}_{22}\tilde{M}_{2}P_{22}P^{T}_{22},\ \ \forall \tilde{M_{2}}\in SR^{(n-k-r_{2})\times (n-k-r_{2})},$

(4.8)

$M_{3}=\tilde{C}_{22}+P_{32}P^{T}_{32}\tilde{M}_{3}P_{32}P^{T}_{32},\ \ \forall \tilde{M_{3}}\in SR^{(k-r_{3})\times (k-r_{3})}.$

(4.9)

Substituting (4.8), (4.9) into (4.5), then we get that the unique solution to problem (1.2) can be expressed in (4.4). The proof is completed.