Throughout this paper, let $R^{n\times m}$ be the set of all $n\times m$ real 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 $A$, $A^{T}$, $A^{+}$, $\|A\|$ and $r(A)$ represent its transpose, Moore-Penrose inverse, Frobenius norm and rank, respectively.

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

Centro-symmetric matrices have practical applications in information theory, linear system theory, linear estimate theory, and numerical analysis(see, e.g. [1-4]).

We know that investigating minimal ranks of matrix expressions has many immediate motivations in matrix analysis and applications. For example, the classical matrix equation $AX=B$ is consistent if and only if

$\begin{eqnarray*} \min\limits_{X}\ {\rm rank}(B-AX)=0. \end{eqnarray*}$

The two consistent matrix equations $A_{1}X_{1}B_{1}=C_{1}$, $A_{2}X_{2}B_{2}=C_{2}$ where $X_{1}$ and $X_{2}$ have the same size, have a common solution if and only if

$\begin{eqnarray*} \min\limits_{X_{1}, X_{2}}\ {\rm rank}(X_{1}-X_{2})=0. \end{eqnarray*}$

In 1972, Mitra [5] considered solutions with fixed ranks for the matrix equations $AX=B$ and $AXB=C$. In 1984, Mitra [6] gave common solutions of minimal rank of the pair of complex matrix equations $AX=C, XB=D$. 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 [7]. In 2003, Tian [8] investigated the extremal rank solutions to the complex matrix equation $AXB=C$ and gave some applications. Xiao et al. [9] in 2009 considered the symmetric minimal rank solution to equation $AX=B$. Recently, the anti-reflexive extremal rank solutions to the matrix equation $AX=B$ was derived by Xiao et al. [10].

In this paper, we consider the centro-symmetric extremal rank solutions of the matrix equation

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

(1.1)

where $A$ and $B$ are given matrices in $R^{m\times n}$.

We also consider the matrix nearness problem

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

(1.2)

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

We organize this paper as follows. In Section 2, we first establish a representation for the centro-symmetric matrix. Then we give necessary and sufficient conditions for the existence of centro-symmetric solution to (1.1). We also give the expressions of such solutions when the solvability conditions are satisfied. We in Section 3 establish formulas of maximal and minimal ranks of centro-symmetric solutions to (1.1), and present the centro-symmetric extremal rank solutions to (1.1). We in Section 4 present the expression of the optimal approximation solution to the set of the minimal rank solution.

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

$\begin{equation} D_{n}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}I_{k}&I_{k}\\S_{k}&-S_{k}\end{array}\right)(n=2k), D_{n}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}I_{k}&0&I_{k}\\0&\sqrt{2}&0\\S_{k}&0&-S_{k}\end{array}\right)(n=2k+1), \end{equation}$

(2.1)

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

Lemma 1 [11] Let $X\in R^{n\times n}$ and $D_{n}$ with the forms of $(2.1)$, then $X$ is the centro-symmetric matrix if and only if there exist $X_{1}\in R^{(n-k)\times (n-k)}$ and $X_{2}\in R^{k\times k}$, whether $n$ is odd or even, such that

$\begin{equation}\label{5} X=D_{n}\left(\begin{array}{cc}X_{1}&0\\0&X_{2}\end{array}\right)D^{T}_{n}. \end{equation}$

(2.2)

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

Given matrices $A_{1}\in R^{m\times n}$, $B_{1}\in R^{m\times p}$, by making generalized singular value decomposition to $[A_{1},B_{1}]$, we have

$\begin{eqnarray} A_{1}=M_{1}\Sigma_{A_{1}}U_{1}, \ \ B_{1}=M_{1}\Sigma_{B_{1}}V_{1}, \end{eqnarray}$

(2.3)

where $M_{1}$ is a $m\times m$ nonsingular matrix, $U_{1}\in OR^{n\times n}$, $V_{1}\in OR^{p\times p}$,

$\begin{eqnarray*} \Sigma_{A_{1}}=\left[\begin{array}{cccc} I & 0 & 0\\ 0 & S_{A_{1}} & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{cccc} r_{1}-s_{1}\\s_{1}\\k_{1}-r_{1}\\m-k_{1} \end{array},\ \ \ \ \Sigma_{B_{1}}=\left[\begin{array}{cccc} 0 & 0 & 0 \\ 0 & S_{B_{1}} & 0 \\0 & 0 & I\\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{cccc} r_{1}-s_{1}\\s_{1}\\k_{1}-r_{1}\\m-k_{1} \end{array}, \end{eqnarray*}$

$k_{1}=r[A_{1},B_{1}]$, $r_{1}=r(A_{1})$, $s_{1}=r(A_{1})+r(B_{1})-r[A_{1},B_{1}]$, $S_{A_{1}}={\rm diag}(\alpha_{1},\cdots,\alpha_{s_{1}})$, $S_{B_{1}}={\rm diag}(\beta_{1},\cdots,\beta_{s_{1}})$, $0<\alpha_{s_{1}}\leq\cdots\leq\alpha_{1}<1$, $0<\beta_{1}\leq\cdots\leq\beta_{s_{1}}<1$, $\alpha^{2}_{i}+\beta^{2}_{i}=1$, $i=1,\cdots,s_{1}$.

Lemma 2 [10] Given matrices $A_{1}\in R^{m\times n}$, $B_{1}\in R^{m\times p}$, the generalized singular ecomposition of the matrix pair $[A_{1},B_{1}]$ is given by $(2.3)$, then matrix equation $A_{1}X=B_{1}$ is consistent, if and only if

$\begin{eqnarray} r[A_{1},B_{1}]=r(A_{1}), \end{eqnarray}$

(2.4)

and the expression of its general solution is

$\begin{equation} X=U_{1}^{T}\left[\begin{array}{ccc}0 & 0 \\ 0 & S_{A_{1}}^{-1}S_{B_{1}} \\ Y_{31} & Y_{32} \\ \end{array}\right]V_{1}, \end{equation}$

(2.5)

where $Y_{31}\in R^{(n-r_{1})\times (p-s_{1})}$, $Y_{32}\in R^{(n-r_{1})\times s_{1}}$ are arbitrary.

Assume $D_{n}$ with the form of $(2.1)$, and $AD_{n}$ and $BD_{n}$ have the following partition form

$\begin{eqnarray} AD_{n}=[A_{2}, A_{3}],\ \ BD_{n}=[B_{2}, B_{3}], \end{eqnarray}$

(2.6)

where $A_{2}\in R^{m\times (n-k)}$, $A_{3}\in R^{m\times k}$, $B_{2}\in R^{m\times (n-k)}$, $B_{3}\in R^{m\times k}$, and the generalized singular value decomposition of the matrix pair $[A_{2} , B_{2}]$, $[A_{3} , B_{3}]$ are, respectively,

$\begin{eqnarray} && A_{2}=M_{2}\Sigma_{A_{2}}U_{2},\ \ B_{2}=M_{2}\Sigma_{B_{2}}V_{2}, \end{eqnarray}$

(2.7)

$\begin{eqnarray} && A_{3}=M_{3}\Sigma_{A_{3}}U_{3}, \ \ B_{3}=M_{3}\Sigma_{B_{3}}V_{3}, \end{eqnarray}$

(2.8)

where $U_{2}\in OR^{(n-k)\times (n-k)}$, $V_{2}\in OR^{(n-k)\times (n-k)}$, $U_{3}\in OR^{k\times k}$, $V_{3}\in OR^{k\times k}$, nonsingular matrices $M_{2}, M_{3}\in R^{m\times m}$, $k_{2}=r[A_{2},B_{2}]$, $r_{2}=r(A_{2})$, $s_{2}=r(A_{2})+r(B_{2})-r[A_{2}, B_{2}]$, and $k_{3}=r[A_{3},B_{3}]$, $r_{3}=r(A_{3})$, $s_{3}=r(A_{3})+r(B_{3})-r[A_{3},B_{3}]$,

$\begin{eqnarray*}&& \Sigma_{A_{2}}=\left[ \begin{array}{ccc} I & 0 & 0 \\ 0 & S_{A_{2}} & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{ccc} r_{2}-s_{2}\\s_{2}\\k_{2}-r_{2}\\m-k_{2} \end{array}, \Sigma_{B_{2}}=\left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & S_{B_{2}} & 0\\ 0 & 0 & I \\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{ccc} r_{2}-s_{2}\\s_{2}\\k_{2}-r_{2}\\m-k_{2} \end{array}, \\ && \Sigma_{A_{3}}=\left[ \begin{array}{ccc} I & 0 & 0 \\ 0 & S_{A_{3}} & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{ccc} r_{3}-s_{3}\\s_{3}\\k_{3}-r_{3}\\m-k_{3} \end{array}, \Sigma_{B_{3}}=\left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & S_{B_{3}} & 0 \\ 0 & 0 & I \\ 0 & 0 & 0\\ \end{array}\right] \begin{array}{ccc} r_{3}-s_{3}\\s_{3}\\k_{3}-r_{3}\\m-k_{3} \end{array}.\end{eqnarray*}$

Then we can establish the existence theorems as follows.

Theorem 1 Let $A,B\in R^{m\times n}$ and $D_{n}$ with the form of $(2.1)$, $AD_{n},BD_{n}$ have the partition forms of $(2.6)$, and the generalized singular value decompositions of the matrix pair $[A_{2},B_{2}]$ and $[A_{3},B_{3}]$ are given by $(2.7)$ and $(2.8)$. Then equation $(1.1)$ has a centro-symmetric solution $X$ if and only if

$\begin{eqnarray} r[A_{2},B_{2}]=r(A_{2}),\ \ r[A_{3},B_{3}]=r(A_{3}), \end{eqnarray}$

(2.9)

and its general solution can be expressed as

$\begin{eqnarray} X=D_{n}\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]V_{2}&0\\0&U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right]V_{3}\end{array}\right]D_{n}^{T}, \end{eqnarray}$

(2.10)

where $Z_{31}\in R^{(n-k-r_{2})\times (n-k-s_{2})}$, $Z_{32}\in R^{(n-k-r_{2})\times s_{2}}$, $W_{31}\in R^{(k-r_{3})\times (k-s_{3})}$, $W_{32}\in R^{(k-r_{3})\times s_{3}}$ are arbitrary.

Proof Suppose the matrix equation (1.1) has a solution $X$ is centro-symmetric, then it follows from Lemma 1 that there exist $X_{1} \in R^{(n-k)\times (n-k)}$, $X_{2} \in R^{k\times k}$ satisfying

$\begin{eqnarray} X=D_{n}\left[\begin{array}{cc}X_{1}&0\\0&X_{2}\end{array}\right]D_{n}^{T} \ \ \ {\rm and} \ \ \ AX=B. \end{eqnarray}$

(2.11)

By (2.6), that is

$\begin{eqnarray} [A_{2}\ \ A_{3}]\left[\begin{array}{cc}X_{1}&0\\0&X_{2}\end{array}\right]=[B_{2}\ \ B_{3}], \end{eqnarray}$

(2.12)

i.e.,

$\begin{eqnarray} A_{2}X_{1}=B_{2},\ \ A_{3}X_{2}=B_{3}. \end{eqnarray}$

(2.13)

Therefore by Lemma 2, (2.9) hold, and

$\begin{eqnarray} X_{1}=U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]V_{2}, \ \ X_{2}=U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right]V_{3}, \end{eqnarray}$

(2.14)

where $Z_{31}\in R^{(n-k-r_{2})\times (n-k-s_{2})}$, $Z_{32}\in R^{(n-k-r_{2})\times s_{2}}$, $W_{31}\in R^{(k-r_{3})\times (k-s_{3})}$, $W_{32}\in R^{(k-r_{3})\times s_{3}}$ are arbitrary. Substituting (2.14) into (2.11) yields that the centro-symmetric solution $X$ of the matrix equation (1.1) can be represented by (2.10). The proof is completed.

Theorem 2 Suppose that the matrix equation $(1.1)$ has a centro-symmetric solution $X$ and $\Omega$ is the set of all centro-symmetric solutions of $(1.1)$. Then the extreme ranks of $X$ are as follows:

(1) The maximal rank of $X$ is

$\begin{eqnarray} \max\limits_{X\in \Omega}r(X)=\min\{n-k,n-k-r(A_{2})+r(B_{2})\}+\min\{k,k-r(A_{3})+r(B_{3})\}. \end{eqnarray}$

(3.1)

The general expression of $X$ satisfying $(3.1)$ is

$\begin{eqnarray} X=D_{n}\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]V_{2}&0\\0&U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right]V_{3}\end{array}\right]D_{n}^{T}, \end{eqnarray}$

(3.2)

where $Z_{31}\in R^{(n-k-r_{2})\times (n-k-s_{2})}$, $W_{31}\in R^{(k-r_{3})\times (k-s_{3})}$ are chosen such that $r(Z_{31})=\min(n-k-r_{2}, n-k-s_{2})$, $r(W_{31})=\min(k-r_{3}, k-s_{3})$, $Z_{32}\in R^{(n-k-r_{2})\times s_{2}}$, $W_{32}\in R^{(k-r_{3})\times s_{3}}$ are arbitrary.

(2) The minimal rank of $X$ is

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

(3.3)

The general expression of $X$ satisfying $(3.3)$ is

$\begin{eqnarray} X=D_{n}\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\0&Z_{32}\end{array}\right]V_{2}&0\\0&U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\0&W_{32}\end{array}\right]V_{3}\end{array}\right]D_{n}^{T}, \end{eqnarray}$

(3.4)

where $Z_{32}\in R^{(n-k-r_{2})\times s_{2}}$, $W_{32}\in R^{(k-r_{3})\times s_{3}}$ are arbitrary.

Proof $(1)$ By (2.10),

$\begin{eqnarray} && \max\limits_{X\in \Omega}r(X)=\max\limits_{Z_{31}}r\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]+\max\limits_{W_{31}}r\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right], \end{eqnarray}$

(3.5)

$\begin{eqnarray} &&\max\limits_{Z_{31}}r\left[\begin{array}{ccc}0&0\\0&S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]\nonumber\\ &=& s_{2}+\min\{n-k-r_{2},n-k-s_{2}\} \nonumber\\ &=& \min\{n-k,n-k-r_{2}+s_{2}\}=\min\{n-k,n-k-r(A_{2})+r(B_{2})\}, \end{eqnarray}$

(3.6)

and

$\begin{eqnarray} && \max\limits_{W_{31}}r\left[\begin{array}{ccc}0&0\\0&S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right] = s_{3}+\min\{k-r_{3},k-s_{3}\} \nonumber\\ &=& \min\{k,k-r_{3}+s_{3}\}= \min\{k,k-r(A_{3})+r(B_{3})\}. \end{eqnarray}$

(3.7)

Taking (3.6) and (3.7) into (3.5) yields (3.1).

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

$(2)$ By (2.10),

$\begin{eqnarray} && \min\limits_{X\in \Omega}r(X)=\min\limits_{Z_{31}}r\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]+\min\limits_{W_{31}}r\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right], \end{eqnarray}$

(3.8)

$\begin{eqnarray} && \min\limits_{Z_{31}}r\left[\begin{array}{ccc}0&0\\0&S_{A_{2}}^{-1}S_{B_{2}}\\Z_{31}&Z_{32}\end{array}\right]=s_{2}=r(B_{2}),\ \ \min\limits_{W_{31}}r\left[\begin{array}{ccc}0&0\\0&S_{A_{3}}^{-1}S_{B_{3}}\\W_{31}&W_{32}\end{array}\right]=s_{3}=r(B_{3}).\nonumber\\ \end{eqnarray}$

(3.9)

Taking (3.9) into (3.8) yields (3.3).

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

From (3.4), when $S_{m}=\{X\mid AX=B, X\in CSR^{n\times n}, r(X)=\min\limits_{Y\in \Omega}r(Y)\}$ is nonempty, it is easy to verify that $S_{m}$ is a closed convex set, therefore there exists a unique solution $\hat{X}$ to the matrix nearness problem (1.2).

Theorem 3 Given matrix $\tilde{X}$, and the other given notations and conditions are the same as in Theorem 1. Let

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

(4.1)

and we denote

$\begin{eqnarray} U_{2}\tilde{X}_{11}V_{2}^{T}=\left[ \begin{array}{ccc} \tilde{Z}_{11} & \tilde{Z}_{12}\\ \tilde{Z}_{21} & \tilde{Z}_{22}\\ \tilde{Z}_{31} & \tilde{Z}_{32}\\ \end{array}\right], \ \ \ U_{3}\tilde{X}_{22}V_{3}^{T}=\left[ \begin{array}{ccc} \tilde{W}_{11} & \tilde{W}_{12}\\ \tilde{W}_{21} & \tilde{W}_{22}\\ \tilde{W}_{31} & \tilde{W}_{32}\\ \end{array}\right]. \end{eqnarray}$

(4.2)

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

$\begin{eqnarray} \hat{X}=D_{n}\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\ 0&\tilde{Z}_{32}\end{array}\right]V_{2}&0\\0& U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\ 0&\tilde{W}_{32}\end{array}\right]V_{3}\end{array}\right]D_{n}^{T}, \end{eqnarray}$

(4.3)

where $\tilde{Z}_{32},\tilde{W}_{32}$ are the same as in $(4.2)$.

Proof  When $S_{m}$ is nonempty, it is easy to verify from (3.4) that $S_{m}$ is a closed convex set. Since $R^{n\times n}$ is a uniformly convex banach space under Frobenius norm, there exists a unique solution for problem (1.2). By theorem 2, for any $X\in S_{m}$, $X$ can be expressed as

$\begin{eqnarray} X=D_{n}\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\0&Z_{32}\end{array}\right]V_{2}&0\\0&U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\0&W_{32}\end{array}\right]V_{3}\end{array}\right]D_{n}^{T}, \end{eqnarray}$

(4.4)

where $Z_{32}\in R^{(n-k-r_{2})\times s_{2}}$, $W_{32}\in R^{(k-r_{3})\times s_{3}}$ are arbitrary.

Using the invariance of the Frobenius norm under unitary transformations, we have

$\begin{aligned} \|X-\tilde{X}\|^2 =& \left\|\left[\begin{array}{cc} U_{2}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{2}}^{-1}S_{B_{2}}\\ 0&Z_{32}\end{array}\right]V_{2}&0\\0& U_{3}^{T}\left[\begin{array}{ccc}0&0\\0& S_{A_{3}}^{-1}S_{B_{3}}\\ 0&W_{32}\end{array}\right]V_{3}\end{array}\right]-D_{n}^{T}\tilde{X}D_{n}\right\|^2\\ =& \left\|Z_{32}-\tilde{Z}_{32}\right\|^2+\left\|W_{32}-\tilde{W}_{32}\right\|^2+\left\|S_{A_{2}}^{-1}S_{B_{2}}-\tilde{Z}_{22}\right\|^2+ \left\|S_{A_{3}}^{-1}S_{B_{3}}-\tilde{W}_{22}\right\|^2\\ &+ \left\|\tilde{X}_{12}\right\|^2+\left\|\tilde{X}_{21}\right\|^2+\left\|\tilde{Z}_{11}\right\|^2+ \left\|\tilde{Z}_{12}\right\|^2+\left\|\tilde{Z}_{21}\right\|^2+\left\|\tilde{Z}_{31}\right\|^2\\ &+ \left\|\tilde{W}_{11}\right\|^2+\left\|\tilde{W}_{12}\right\|^2+\left\|\tilde{W}_{21}\right\|^2+\left\|\tilde{W}_{31}\right\|^2. \end{aligned}$

Therefore, $\|X-\tilde{X}\|$ reaches its minimum if and only if

$\begin{eqnarray} Z_{32}=\tilde{Z}_{32},\ \ W_{32}=\tilde{W}_{32}. \end{eqnarray}$

(4.5)

Substituting (4.5) into (4.4) yields (4.3). The proof is completed.