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.