## The Properties and Applications of the MP Weak Core Inverse

Liu Xiaoji,, Liao Mengyue,, Jin Hongwei,

School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006

 Fund supported: the National Natural Science Foundation of China.  12061015the Guangxi Natural Science Foundation.  2018GXNSFDA281023the Special Fund for Science and Technological Bases and Talents of Guangxi.  桂科AD21220024

Abstract

In this paper, the concept of the Moore-Penrose weak Core inverse (MPWC inverse) is proposed based on the Moore-Penrose inverse and the weak Core inverse. It is described from algebraic and geometric perspectives respectively. The relationship between the MP weak Core inverse and the nonsingular bordered matrix is given. The expression of the MP weak Core inverse is given by using the Hartwig-Spindelböck decomposition and the Core-EP decomposition. The equivalence between the MP weak Core inverse of a matrix and EP matrix, the characterization and the perturbation analysis are given.

Keywords： Weak Core inverse ; Moore-Penrose inverse ; MPWC inverse ; Core-EP decomposition

Liu Xiaoji, Liao Mengyue, Jin Hongwei. The Properties and Applications of the MP Weak Core Inverse. Acta Mathematica Scientia[J], 2022, 42(6): 1619-1632 doi:

## 1 引言

${\mathbb C}^{m\times n}$表示$m\times n$阶全体复矩阵的集合. 记$A^\ast$, $R(A)$, $N(A)$, ${\rm rk}(A)$, $\|A\|$分别表示矩阵$A$的共轭转置, 值域, 零空间, 秩, 谱范数. 对于$A\in{\mathbb C}^{m\times n}$, 如果$X\in{\mathbb C}^{n\times m}$满足$AXA=A$, $XAX=X$, $(AX)^\ast=AX$, $(XA)^\ast=XA$, 则$X$称为$A$的Moore-Penrose逆[45]. 矩阵$A$的Moore-Penrose逆是唯一的, 记为$A^\dagger$.$P_{A}=AA^\dagger$, $Q_{A}=A^\dagger A$. 此外, 如果$A$满足$AA^\dagger=A^\dagger A$, 或者$R(A)=R(A^\ast)$, 我们称$A$是EP矩阵. 更多关于Moore-Penrose逆的性质, 可以参考文献[610]. 如果$X$满足$XAX=X$, 则称之为$A$的外逆, 用$A^{(2)}$表示. $X$满足$AXA=A$, $XAX=X$, 则称$X $$A 的自反广义逆. 一个矩阵 X 满足 XAX=X , R(A)=T , N(A)=S , 则 X 是唯一的, 记为 A_{T, S}^{(2)} . 矩阵 A 的指标记为 {\mbox{Ind}}(A)=k , 其中 k 是满足 {\rm rk}(A^{k+1})={\rm rk}(A^k) 成立的最小非负整数. 设 A\in{\mathbb C}^{n\times n} , 矩阵 X\in{\mathbb C}^{n\times n} 满足 AXA^k=A^k , XAX=X , AX=XA , 则称之为 A 的Drazin逆, 用 A^D 表示[11]. 如果 {\mbox{Ind}}(A)\leq1 , 则称之为 A 的群逆, 记为 A^\sharp . 2010年, Baksalary和Trenkler[12]提出Core逆的概念: 设 A\in{\mathbb C}^{n\times n} , {\mbox{Ind}}(A)\leq1 , 则存在唯一的矩阵 X\in{\mathbb C}^{n\times n} 满足 AX=AA^\dagger$$ R(X)\subseteq R(A)$, 称之为$A$的Core逆, 记作. 更多关于Core逆的刻画和应用参考文献[1314]. Malik和Thom[15]引入了一种广义的Core逆: DMP逆. 设$A\in{\mathbb C}^{n\times n} $${\mbox{Ind}}(A)=k , 则存在唯一的矩阵 X 满足 XAX=X , XA=A^DA , A^kX=A^kA^\dagger , 称 X$$ A$的DMP逆, 记之为$A^{D, \dagger}$, 且$A^{D, \dagger}=A^DAA^\dagger$. Manjunatha Prasad和Mohana[16]引入Core-EP逆的概念: 设$A\in{\mathbb C}^{n\times n} $${\mbox{Ind}}(A)=k , 则存在唯一的矩阵 X 满足 XAX=X , R(X)\subseteq R(X^\ast)\subseteq R(A^k) , 称之为 A 的Core-EP逆, 记为. 而且Core-EP逆可以表示为. Mehdipour和Salemi在文献[17]中引入了 A 的CMP逆的概念: 设 A\in{\mathbb C}^{n\times n}$$ {\mbox{Ind}}(A)=k$, 则存在唯一的矩阵$X$满足$XAX=X$, $AXA=A_{1}$, $AX=A_{1}A^\dagger$, $XA=A^\dagger A_{1}$, 其中$A_{1}=AA^DA$, 称之为$A$的CMP逆, 记为$A^{C, \dagger}$. 更多关于DMP逆、Core-EP逆、CMP逆的性质和刻画参考文献[1824]. Wang和Chen[25]利用Core-EP分解提出了弱群逆的概念: 设$A\in{\mathbb C}^{n\times n} $${\mbox{Ind}}(A)=k , 则称唯一满足 AX^2=X , 的矩阵 X$$ A$的弱群逆, 记作$A^{{ⓦ}}$, 且. 文献[2627]研究了弱群逆的相关性质、刻画和应用, 建立了若干偏序和预偏序. 文献[2829]分别将弱群逆推广到一般矩阵和线性算子上. 最近, Ferreyra等人在文献[30]中引入了矩阵弱Core部分的概念和弱Core逆的概念. 记$A$的弱Core部分为$C=AA^{{ⓦ}}A$.$A\in{\mathbb C}^{n\times n} $${\mbox{Ind}}(A)=k , 存在一个唯一的 X 满足 则称 X$$ A$的弱Core逆, 记作$A^{{ⓦ}, \dagger}$, 并给出弱Core逆的刻画$A^{{ⓦ}, \dagger}=A^{{ⓦ}}AA^\dagger$. 进一步, 文献[30]中引入对偶弱Core逆的概念并给出刻画, $A^{\dagger, {ⓦ}}=A^\dagger AA^{{ⓦ}}$. 文献[31]中研究了弱Core逆的性质和表征, 文献[32]将弱Core逆推广到线性算子上. 这些新型广义逆为我们解决新的问题提供了新的工具.

## 2 MPWC逆

$(1)$因为$A^\circ=A^\dagger AA^{{ⓦ}}AA^\dagger$, $A^{{ⓦ}}AA^{{ⓦ}}=A^{{ⓦ}}$, 有

$(2)$因为

$(3)$因为$A^\circ A=A^\dagger A^DA^k(A^k)^\dagger A^2$, 所以

$(1) $$X=A^\circ ; (2)$$ A^\dagger CX=X$, $AX=CA^\dagger$, $XA=A^\dagger C$;

$(3) $$XCX=X , CX=CA^\dagger , XC=A^\dagger C ; (4)$$ XCX=X$, $A^{{ⓦ}}AX=A^{{ⓦ}}AA^\dagger$, $XAA^{{ⓦ}}=A^\dagger AA^{{ⓦ}}$.

$(1)\Rightarrow (2) $$X=A^\circ$$ XAX=X$, $AX=CA^\dagger$, $XA=A^\dagger C$, 于是$A^\dagger CX=XAX=X$.

$(2)\Rightarrow (1) $$XA=A^\dagger C$$ XAX=A^\dagger CX=X$. 于是$X=A^\circ$.

$(1)\Rightarrow (3)$由于$X=A^\circ=A^\dagger AA^{{ⓦ}}AA^\dagger$, 则

$(3)\Rightarrow (1) $$CX=CA^\dagger , XC=A^\dagger C , 有 X=XCX=A^\dagger CX=A^\dagger CA^\dagger=A^\circ . (3)\Rightarrow (4)$$ CX=CA^\dagger$左乘$A^{{ⓦ}}$, 得到$A^{{ⓦ}}AA^{{ⓦ}}AX =A^{{ⓦ}}AA^{{ⓦ}}AA^\dagger$, 于是$A^{{ⓦ}}AX=A^{{ⓦ}}AA^\dagger$, 同理$XC=A^\dagger C$右乘$A^{{ⓦ}}$, 有$XAA^{{ⓦ}}AA^{{ⓦ}} =A^\dagger AA^{{ⓦ}}AA^{{ⓦ}}$, 于是$XAA^{{ⓦ}}=A^\dagger AA^{{ⓦ}}$.

$(4)\Rightarrow (3) $$A^{{ⓦ}}AX=A^{{ⓦ}}AA^\dagger 左乘 A , 可以得到 CX=CA^\dagger , 由 XAA^{{ⓦ}}=A^\dagger AA^{{ⓦ}} 右乘 A , 有 XC=A^\dagger C . 证毕. 注2.3 观察上面定理可以发现 A^\circ$$ A$的外逆且$A^\circ $$C 的自反广义逆. 下面, 我们研究 (B, C) 逆和MPWC逆之间的联系, 我们证明矩阵 A\in{\mathbb C}^{n\times n} 的MPWC逆是 A$$ (A^\dagger CA^\ast, A^\ast CA^\dagger)$逆. 首先, 我们给出$(B, C)$逆的定义.

$(2) $$AA^\circ=AA^\dagger ; (3)$$ A^\circ A=A^\dagger A$;

$(4) $$A=AA^{{ⓦ}}A ; (5)$$ A=AA^\circ A$.

$(1)\Leftrightarrow(2) $$A^\circ=A^\dagger 左乘 A 可得 AA^\circ=AA^\dagger , 又由 AA^\circ=AA^\dagger 左乘 A^\dagger , 可得 A^\dagger AA^\circ=A^\dagger AA^\dagger , 即 A^\circ=A^\dagger . (1)\Leftrightarrow(3)$$ A^\circ=A^\dagger$右乘$A$可得$A^\circ A=A^\dagger A$, 又由$A^\circ A=A^\dagger A$右乘$A^\dagger$, 有$A^\circ AA^\dagger=A^\dagger AA^\dagger$可得$A^\circ=A^\dagger$.

$(1)\Leftrightarrow(4) $$A^\circ=A^\dagger 两边同乘 A 可得 A=AA^{{ⓦ}}A , 由 A=AA^{{ⓦ}}A 两边同乘 A^\dagger 可得 A^\circ=A^\dagger . (4)\Leftrightarrow(5) 因为 AA^\circ A= AA^\dagger AA^{{ⓦ}}AA^\dagger A= AA^{{ⓦ}}A , 所以结论成立. 证毕. 众所周知, 广义逆可以表示为具有给定值域和零空间的一种特殊的外逆. 接下来, 我们将给出MPWC逆的类似刻画. 定理2.7 设 A\in{\mathbb C}^{n\times n} 且有 {\mbox{Ind}}(A)=k , 则 由于 A^\circ$$ A$的一个外逆, 且

$R(A)=R(A^\dagger A^k)$. 又由文献[30, 定理3.16]有

$$$x_{ij}= \left.\det\left(\begin{array}{cccccc} A(i\rightarrow d_{j}) \; & B \\ C(i\rightarrow 0) \; & 0 \\ \end{array}\right) \right/ \det\left(\begin{array}{cccccc} A \; & B \\ C \; & 0 \\ \end{array}\right) , i=1, 2, \cdots , n, j=1, 2, \cdots , m,$$$

由于$X$是约束矩阵方程(2.4)的解, 有$R(X)\subseteq R(A^\dagger A^k)=N(C)$, 于是$CX=0$

## 3 MPWC逆的两种标准型及其应用

$$$A=U\left(\begin{array}{cccccc} \Sigma K & \; \Sigma L \\ 0 & \; 0 \\ \end{array}\right) U^\ast,$$$

$$$A_{1}=U\left(\begin{array}{cccccc} T \; & S \\ 0 \; & 0 \\ \end{array}\right) U^\ast, {\quad} A_{2}=U\left(\begin{array}{cccccc} 0\; & 0 \\ 0 \; & N \\ \end{array}\right) U^\ast,$$$

$U\in{\mathbb C}^{n\times n}$是酉矩阵, $T$是非奇异矩阵, ${\rm rk}(T)=r$, $N $$k 阶幂零的. 且 $$A^\circ=U\left(\begin{array}{cccccc} T^\ast\bigtriangleup \; & T^\ast\bigtriangleup T^{-1}SP_{N} \\ (I-Q_{N})S^\ast\bigtriangleup \; & (I-Q_{N})S^\ast\bigtriangleup T^{-1}SP_{N} \\ \end{array}\right) U^\ast,$$ 其中 \bigtriangleup=[TT^\ast+S(I-Q_{N})S^\ast]^{-1} . 众所周知, 如果 A 是一个非奇异矩阵, 则 X=A^{-1} 是下述秩等式的唯一解, 为了得到MPWC逆的一个类似表示形式, 我们对奇异矩阵 A 给出了下面结论. 首先给出下面引理. 引理3.3[34] 设 A\in{\mathbb C}^{n\times n} , M= \left(\begin{array}{cccccc} A \; & AU \\ VA \; & B \\ \end{array}\right) \in{\mathbb C}^{2n\times 2n} , 则 定理3.2 设 A\in{\mathbb C}^{n\times n} , 有 {\mbox{Ind}}(A)=k , {\rm rk}(A^k)=r , 则存在唯一的矩阵 X 满足 $$XA^k=0, X^2=X, (A^k)^\ast AP_{A}X=0, {\rm rk}(X)=n-r;$$ 存在唯一的矩阵 Y 满足 $$YA^\dagger A^k=0, Y^2=Y, (A^k)^\ast A^2Y=0, {\rm rk}(Y)=n-r;$$ 和唯一的矩阵 Z 满足 $${\rm rk}\left(\begin{array}{cccccc} A \; & I-X \\ I-Y \; & Z \\ \end{array}\right) ={\rm rk}(A) .$$ 此时, 有 Z=A^\circ , X=I-AA^\circ , Y=I-A^\circ A . 假设 A 有形式(3.8)式, 容易验证分块矩阵 满足(3.10)式中所有等式. 下面证明解的唯一性. 设 X_{0} 也满足(3.10)式中等式. 设 X_{1}=U^\ast X_{0}U 有分块形式 X_{1}=\left(\begin{array}{cccccc} D_{1} \; & D_{2} \\ D_{3} \; & D_{4} \\ \end{array}\right) , 其中 D_{1}$$ r\times r$的块. 由$XA^k=0$, 以及$T$是可逆的, 可以得到

$T$的非奇异性有$F_{1}T^\ast+F_{2}(I-Q_{N})S^\ast=0$, $F_{3}T^\ast+F_{4}(I-Q_{N})S^\ast=0$, 即

$$$F_{1}=-F_{2}(I-Q_{N})S^\ast(T^\ast)^{-1}, F_{3}=-F_{4}(I-Q_{N})S^\ast(T^\ast)^{-1}.$$$

$$$F_{1}=T^{-1}(S+T^{-1}SN)F_{3}, F_{2}=T^{-1}(S+T^{-1}SN)F_{4}.$$$

设$A$有形式(3.8), $E= U\left(\begin{array}{cccccc} E_{1} &\; E_{2} \\ E_{3} &\; E_{4} \\ \end{array}\right)U^\ast$, 其中$E_{1}\in{\mathbb C}^{r\times r}$. 由(3.9)式, 我们有

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