2 Preliminaries
In this paper, we need the following concepts and conclusions:
Definition 1 [12] For\(A=\left(a_{\text{ij}}\right)\in C^{m\times n},B=\left(b_{\text{ij}}\right)\in C^{p\times q}\), called the following block matrix
\begin{equation} A\otimes B=\left(a_{\text{ij}}B\right)\in C^{mp\times nq}\nonumber \\ \end{equation}
is the Kronecker product of matrix \(A\) and matrix \(B\).
Definition 2 [1]For\(\ A=A_{1}+A_{2}i\in C^{m\times n}\), which\(A_{1},A_{2}\in R^{m\times n}\).
\(A^{R}=\par \begin{pmatrix}A_{1}&-A_{2}\\ A_{2}&A_{1}\\ \end{pmatrix}\in R^{2m\times 2n}\),
is called to the real representation matrix of \(A\), Let\(\text{\ \ }A_{c}^{R}\) denote the first column block\(\par \begin{pmatrix}A_{1}\\ A_{2}\\ \end{pmatrix}\ \)of \(A^{R}\).
\(A^{R}\) and \(A_{c}^{R}\) have some properties as follows:
Lemma 1 [1] For\(A,B\in C^{m\times n},k\in C.\) Then the following properties hold.
(1) \(A=B\Leftrightarrow A^{R}=B^{R}\); (2)\(\left(A+B\right)^{R}A^{R}+B^{R}\); (3)\(\left(\text{kA}\ \right)^{\text{R\ }}=kA^{R}\).
Lemma 2 [1]For\(\ A,B\in C^{m\times n},D\in C^{n\times s},k\in C.\) Then the following properties hold.
(1) \(A=B\Leftrightarrow A_{c}^{R}=B_{c}^{R}\); (2)\(\left(\text{kA}\right)_{c}^{R}=kA_{c}^{R}\),
(3) \({(AD)}_{c}^{R}=A^{R}D_{c}^{R}\); (4)\(\left(A+B\right)_{c}^{R}A_{c}^{R}+B_{c}^{R}.\)
For\(A\in C^{m\times n},\ have\ \left\|A\right\|=\frac{1}{2}\left\|A^{R}\right\|=\left\|A_{c}^{R}\right\|\text{\ and\ }\).
The vec operator of real matrix have some properties as follows:
Lemma 3[12] For\(A\in R^{m\times n},B\in R^{n\times p},C\in R^{p\times q}\). Then the following properties hold.
\begin{equation} \text{vec}\left(\text{ABC}\right)=\left(C^{T}\otimes A\right)\text{vec}\left(B\right).\nonumber \\ \end{equation}
For complex matrix\(X=X_{0}+X_{1}i\), the following relationship between\(X^{R}X_{C}^{R}{X}_{0}\text{\ and\ }X_{1}\ \)
as follows:
Lemma 4[1] For\(X=X_{0}+X_{1}i\in C^{m\times n}.\) Then the following properties hold. (1)\(\text{vec}\left(X^{R}\right)=\text{Hvec}\left(X_{C}^{R}\right)\),
in which \(H=\par \begin{pmatrix}\text{diag}\left(I_{2n},\ldots,I_{2n}\right)\\ \text{diag}\left(Q_{n},\ldots,Q_{n}\right)\\ \end{pmatrix}\in R^{4mn\times 2\text{mn}}\),\(Q_{n}=\par \begin{pmatrix}0&-I_{n}\\ I_{n}&0\\ \end{pmatrix}.\)
(2) \(\text{vec}{(X}_{c}^{R})=K\begin{pmatrix}\text{vec}X_{0}\\ \text{vec}X_{1}\\ \end{pmatrix}\),
in which \(K=\par \begin{pmatrix}\par \begin{matrix}I_{n}&0\\ 0&0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&0\\ \end{matrix}&0\\ \par \begin{matrix}\cdots&0\\ \end{matrix}&I_{n}\\ \end{matrix}&\par \begin{matrix}0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ 0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}0&I_{n}\\ 0&0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&0\\ \end{matrix}&0\\ \par \begin{matrix}\cdots&0\\ \end{matrix}&0\\ \end{matrix}&\par \begin{matrix}0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ I_{n}&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}\cdots&\cdots\\ \par \begin{matrix}0\\ 0\\ \end{matrix}&\par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&\cdots\\ \end{matrix}&\cdots\\ \par \begin{matrix}\par \begin{matrix}\cdots&I_{n}\\ \end{matrix}\\ \par \begin{matrix}\cdots&0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\cdots&\par \begin{matrix}\cdots&\cdots\\ \end{matrix}\\ \par \begin{matrix}0\\ 0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \par \begin{matrix}\cdots&I_{n}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{pmatrix}\in C^{{2n}^{2}\times 2n^{2}}.\)
(3) if\(\ X=X_{0},\) have
\(\text{vec}\left(X_{c}^{R}\right)=Vvec\left(X_{0}\right)\),
in which\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }V=\par \begin{pmatrix}I_{n}&0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ 0&0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \par \begin{matrix}0\\ 0\\ \par \begin{matrix}\cdots\\ \par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}I_{n}\\ 0\\ \par \begin{matrix}\cdots\\ 0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \par \begin{matrix}\cdots&0\\ \end{matrix}\\ \par \begin{matrix}\par \begin{matrix}\cdots\\ \cdots\\ \cdots\\ \end{matrix}&\par \begin{matrix}\cdots\\ I_{n}\\ 0\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{pmatrix}\in C^{{2n}^{2}\times n^{2}}.\)
By calculation, give definition and relationship between the independent elements of real (dis)symmetric matrix and vec of its independent elements:
Definition 3 [1] For\(X=\left(x_{\text{ij}}\right)\in\text{SR}^{n\times n}\), let\({\alpha_{1}=(x}_{11},x_{21,}\ldots x_{n1}),{\alpha_{2}=(x}_{22},\ x_{32,}\)
\(\ldots x_{n2}),\ {\ldots,\alpha_{n-1}=(x}_{(n-1)(n-1)},x_{n(n-1)}),\alpha_{n}=x_{\text{nn}}\), the elements of \(\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\) are called as the independent elements of real symmetric matrix\(X=\left(x_{\text{ij}}\right)\in\text{SR}^{n\times n}\), for short independent entry. Denoted by
\(\text{\ \ }\text{vec}_{S}\left(X\right)=\left(\alpha_{1},\alpha_{2,}\ldots\alpha_{n}\right)^{T}\in R^{\frac{n(n+1)}{2}}\)
is called a column straight of the independent elements of real symmetric matrix\(X=\left(x_{\text{ij}}\right)\in\text{SR}^{n\times n}.\)
Definition 4 [1] For arbitrary real dissymmetric matrix \(X=(x_{\text{ij}})\in\text{ASR}^{n\times n}\), and let\({\beta_{1}=(x}_{21},x_{31,}\ldots x_{n1}),\ {\beta_{2}=(x}_{32},x_{42,}\ldots x_{n2}),\ldots,{\beta_{n-1}=(x}_{(n-1)(n-2)},x_{n(n-2)}),\beta_{n}=x_{n(n-1)}\), the elements of \(\beta_{1},\beta_{2},\ldots,\beta_{n}\) are called as the independent elements of real dissymmetric matrix dissymmetric matrix\(X=(x_{\text{ij}})\in\text{ASR}^{n\times n}\). Denoted by
\begin{equation} \text{ves}_{A}\left(X\right)=\left(\beta_{1},\beta_{2,}\ldots\beta_{n}\right)^{T}\in R^{\frac{n(n-1)}{2}}\nonumber \\ \end{equation}
is called a column straight of the independent elements of real dissymmetric matrix
Lemma 5 [1]For\(\ X=\left(x_{\text{ij}}\right)\in\text{SR}^{n\times n}\). Then
\(X\in\text{SR}^{n\times n}\Leftrightarrow\text{ves}\left(X\right)=K_{n}\text{ves}_{S}\left(X\right),K_{n}\in R^{n^{2}\times\frac{n(n+1)}{2}}\),
in which \(K_{n}=\left(\par \begin{matrix}e_{1}&e_{2}&\par \begin{matrix}e_{3}&\cdots&e_{n-1}\\ \end{matrix}\\ 0&e_{1}&\par \begin{matrix}0&\cdots&0\\ \end{matrix}\\ \par \begin{matrix}0\\ \cdots\\ \par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ 0\\ 0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\par \begin{matrix}e_{1}\\ \cdots\\ \end{matrix}&\par \begin{matrix}\cdots\\ \\ \end{matrix}&\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}0&\cdots&e_{1}\\ \end{matrix}\\ \par \begin{matrix}0&\cdots&0\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\par \begin{matrix}e_{n}&0&\par \begin{matrix}0&\cdots&0\\ \end{matrix}\\ 0&e_{2}&\par \begin{matrix}e_{3}&\cdots&e_{n-1}\\ \end{matrix}\\ \par \begin{matrix}0\\ \cdots\\ \par \begin{matrix}0\\ e_{1}\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ 0\\ 0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}&\par \begin{matrix}\cdots\\ \\ \end{matrix}&\par \begin{matrix}0\\ \vdots\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}0&\cdots&e_{2}\\ \end{matrix}\\ \par \begin{matrix}0&\cdots&0\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\par \begin{matrix}0&\cdots&\par \begin{matrix}0&0&0\\ \end{matrix}\\ e_{n}&\cdots&\par \begin{matrix}0&0&0\\ \end{matrix}\\ \par \begin{matrix}0\\ \vdots\\ \par \begin{matrix}0\\ e_{2}\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots\\ \\ \end{matrix}\\ \cdots\\ \cdots\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}&\par \begin{matrix}0\\ \cdots\\ \end{matrix}&\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}0&e_{n}&0\\ \end{matrix}\\ \par \begin{matrix}0&e_{n-1}&e_{n}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\right).\)
Lemma 6 [1] For\(X=\left(x_{\text{ij}}\right)\in R^{n\times n}\). Then
\(X\in\text{ASR}^{n\times n}\Leftrightarrow\text{ves}\left(X\right)=L_{n}\text{ves}_{A}\left(X\right),L_{n}\in R^{n^{2}\times\frac{n(n+1)}{2}}\),
in which \(L_{n}=\left(\par \begin{matrix}e_{2}&e_{3}&\par \begin{matrix}\cdots&e_{n-1}&e_{n}\\ \end{matrix}\\ {-e}_{1}&0&\par \begin{matrix}\cdots&0&0\\ \end{matrix}\\ \par \begin{matrix}0\\ \cdots\\ \par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}{-e}_{1}\\ \cdots\\ \end{matrix}\\ 0\\ 0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\par \begin{matrix}\cdots\\ \\ \end{matrix}&\par \begin{matrix}0\\ \\ \end{matrix}&\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}\cdots&{-e}_{1}&0\\ \end{matrix}\\ \par \begin{matrix}\cdots&0&{-e}_{1}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\par \begin{matrix}0&\cdots&\par \begin{matrix}0&0&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \end{matrix}\\ e_{3}&\cdots&\par \begin{matrix}e_{n-1}&e_{n}&\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}{-e}_{2}\\ \cdots\\ \par \begin{matrix}0\\ 0\\ \end{matrix}\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}\\ 0\\ 0\\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\par \begin{matrix}0\\ \cdots\\ \end{matrix}&\par \begin{matrix}0\\ \\ \end{matrix}&\par \begin{matrix}\par \begin{matrix}\cdots&0\\ \end{matrix}\\ \par \begin{matrix}&\cdots\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}{-e}_{2}&0&\par \begin{matrix}\cdots&e_{n}\\ \end{matrix}\\ \end{matrix}\\ \par \begin{matrix}0&\par \begin{matrix}{-e}_{2}&\par \begin{matrix}&{-e}_{n-1}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\\ \end{matrix}\right)\)
Lemma 7 [1] Let\(A\in R^{m\times n},\ b\in R^{n}\). Then the least square solutions of the incompatible linear equations \(Ax=b\) is\(\ x=A^{+}b+(I-AA^{+})y\), where \(y\in R^{n}\) is arbitrary vector.
2 The minimum norm least square solution of complex matrix equations\(\left(\mathbf{AXB,CXD}\right)\mathbf{=}\left(\mathbf{E,F}\right)\)
In this paper, the special structure of the real representation matrix of the complex matrix and the MP inverse property of the matrix, we study Hermitian least square solutions of complex matrix equations(1), and give the expressions of Hermitian minimum norm least square solution, real symmetric minimum norm least square solution and real dissymmetric minimum norm least square solution.
Theorem 1 Let\(A,C\in C^{m\times n},B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},{X=X}_{0}+X_{1}i,\)where\(X_{0}\in\text{SR}^{n\times n},X_{1}\in\text{ASR}^{n\times n}\). For complex matrix equations (1), the Hermitian minimum norm least square solution \(X\) satisfies
\(\left(\frac{\text{ves}_{S}\left(X_{0}\right)}{\left(\text{ves}_{A}\left(X_{1}\right)\right)}\right)=\left[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HV}K_{n}\right]^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\).
Proof. By lemma 1-6, we get
\begin{equation} \text{\ \ \ \ \ \ \ \ \ \ \ \ }\left\|\text{AXB}-E\right\|+\left\|\text{CXD}-F\right\|\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|A^{R}\text{\ X}^{R}B_{c}^{R}-E_{c}^{R}\right\|+\left\|C^{R}\text{\ X}^{R}D_{c}^{R}-F_{c}^{R}\right\|\text{\ \ \ }\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|\left(\left(B_{c}^{R}\right)^{T}\otimes A^{R}\right)\text{vec}\left(\text{\ X}^{R}\right)-vec\ (E_{c}^{R})\right\|+\left\|\left(\left(D_{c}^{R}\right)^{T}\otimes C^{R}\right)\text{vec}\left(\text{\ X}^{R}\right)-vec\ {(F}_{c}^{R})\right\|\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)\text{vec}\left({\ X}^{R}\right)-vec(\ E_{c}^{R})}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)\text{vec}\left({\ X}^{R}\right)-vec(\ F_{c}^{R})}\right)\right\|\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{vec}{\ (X}^{R})-\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\right\|\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HK}\begin{pmatrix}\text{vec}X_{0}\\ \text{vec}X_{1}\\ \end{pmatrix}-\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\right\|\nonumber \\ \end{equation}\begin{equation} \ \ \ \ =\left\|\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HK}\begin{pmatrix}K_{n}\text{ves}_{S}\left(X_{0}\right)\\ L_{n}\text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}-\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\right\|\nonumber \\ \end{equation}
Where \(H,K,K_{n},L_{n}\) are defined in the form of lemma 4-6. Let\(G=diag\left(K_{n},L_{n}\right),\ \)then
\(\left\|\text{AXB}-E\right\|+\left\|\text{CXD}-F\right\|\ =\left\|\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HKG}\par \begin{pmatrix}\text{ves}_{S}\left(X_{0}\right)\\ \text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}-\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\right\|\),
So calculating the least squares solution of the complex matrix equation (1) is equivalent to calculating the least squares solution of the following real linear system
\(\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HKG}\par \begin{pmatrix}\text{ves}_{S}\left(X_{0}\right)\\ \text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}=\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\).
By lemma 7, the set of Hermitian least square solution of (1) is denoted by
\({C_{H}=\{X|\par \begin{pmatrix}\text{ves}_{S}\left(X_{0}\right)\\ \text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}=[\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)HKG]}^{+}+[\left(I-\left[\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HKG}\right]^{+}\left(\frac{\left({{(B}_{c}^{R})}^{T}\otimes A^{R}\right)}{\left((D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HKG}\right]y\}\),
Where \(y\) is arbitrary vector. In particular, the Hermitian minimum norm least square solution of (1) is can be expressed as\(X=X_{0}+X_{1}i\in C^{n\times n}\), in which \(X_{0}X_{1}\)satisfies
\(\par \begin{pmatrix}\text{ves}_{S}\left(X_{0}\right)\\ \text{ves}_{A}\left(X_{1}\right)\\ \end{pmatrix}={[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)HKG]}^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\)证毕.
Corollary 1 Let\(A,C\in C^{m\times n},B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},{X=X}_{0}+X_{1}i\),in which\(\ X_{1}=0,X_{0}\in\text{SR}^{n\times n}\). For complex matrix equations (1), the real symmetric minimum norm least square solution\(\ X=X_{0}\) satisfies
\(\text{ves}_{S}\left(X_{0}\right)=\left[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HV}K_{n}\right]^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\),
Where \(H,V,\text{\ K}_{n}\) are defined in the form of lemma 4-6.
Corollary 2Let\(\ A,C\in C^{m\times n},B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},{X=X}_{0}+X_{1}i\),in which\(\ X_{1}=0,X_{0}\in\text{ASR}^{n\times n}\). For complex matrix equations (1), the real dissymmetric minimum norm least square solution\(\ X=X_{0}\) satisfies
\(\text{ves}_{A}\left(X_{0}\right)=\left[\left(\frac{{{(B}_{c}^{R})}^{T}\otimes A^{R}}{\left(D_{c}^{R})^{T}\otimes C^{R}\right)}\right)\text{HV}L_{n}\right]^{+}\left(\frac{E_{c}^{R}}{F_{c}^{R}}\right)\),
Where\(\ H,V,\text{\ L}_{n}\) are defined in the form of lemma 4-6.