﻿ MB-矩阵子直和仍为MB-矩阵的条件 Conditions That Subdirect Sums of MB-Matrices Is Still MB-Matrices

Pure Mathematics
Vol.07 No.06(2017), Article ID:22573,9 pages
10.12677/PM.2017.76055

Conditions That Subdirect Sums of MB-Matrices Is Still MB-Matrices

Yi Luo

School of Mathematics and Statistics, Yunnan University, Kunming Yunnan

Received: Oct. 14th, 2017; accepted: Oct. 28th, 2017; published: Nov. 2nd, 2017

ABSTRACT

By splitting an MB-matrix A into a sum of a nonsingular M-matrix and a nonnegative rank 1 matrix, some sufficient and necessary conditions and some sufficient conditions are given such that the subdirect sum of two MB-matrices is still an MB-matrix. Some examples are also given to illustrate the results.

Keywords:Z-Matrix, Nonsingular M-Matrix, MB-Matrix, Subdirect Sum

MB-矩阵子直和仍为MB-矩阵的条件

1. 引言

1999年Fallat和Johnson首先在文献 [1] 中提出矩阵的子直和的概念，由于其在诸如马可夫链的递增许瓦兹迭代及分裂和重叠的递增许瓦兹迭代等研究中的重要性，引起了学者的关注和研究，并取得了一些重要研究成果，如文献 [2] [3] [4] 分别对非奇异M-矩阵及其逆的子直和、H-矩阵和双对角占优矩阵的子直和等进行了研究。本文在文献 [2] 和 [5] 的基础上对MB-矩阵的子直和进行研究，试图得到MB-矩阵的子直和仍为MB-矩阵的一些新的条件。

2. 预备知识

$A=\left({a}_{ij}\right)\in {R}^{m×n}$ ，如果对于所有的 $i=1,\cdots ,m;j=1,\cdots ,n$ 都有 ${a}_{ij}>0\text{\hspace{0.17em}}\left({a}_{ij}\ge 0\right)$ ，则称 $A$ 为正(非负)矩阵，记为 $A>O\text{\hspace{0.17em}}\left(A\ge O\right)$

${A}^{z}=\left[\begin{array}{cccc}{a}_{11}-{\beta }_{1}^{A}& {a}_{12}-{\beta }_{1}^{A}& \cdots & {a}_{1n}-{\beta }_{1}^{A}\\ {a}_{21}-{\beta }_{2}^{A}& {a}_{22}-{\beta }_{2}^{A}& \cdots & {a}_{2n}-{\beta }_{2}^{A}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}-{\beta }_{n}^{A}& {a}_{n2}-{\beta }_{n}^{A}& \cdots & {a}_{nn}-{\beta }_{n}^{A}\end{array}\right]$${A}^{r}=\left[\begin{array}{cccc}{\beta }_{1}^{A}& {\beta }_{1}^{A}& \cdots & {\beta }_{1}^{A}\\ {\beta }_{2}^{A}& {\beta }_{2}^{A}& \cdots & {\beta }_{2}^{A}\\ ⋮& ⋮& \ddots & ⋮\\ {\beta }_{n}^{A}& {\beta }_{n}^{A}& \cdots & {\beta }_{n}^{A}\end{array}\right]$ (1)

${\beta }_{i}^{A}=\mathrm{max}\left\{0,{a}_{ij}|\forall j\ne i\right\}$ 。显然 ${A}^{z}$ 是Z-矩阵， ${A}^{r}$ 是秩1非负矩阵。若 ${A}^{z}$ 为M-矩阵，则称 $A$ 为MB-矩阵。

$A=\left[\begin{array}{cc}{A}_{11}& {A}_{12}\\ {A}_{21}& {A}_{22}\end{array}\right]$$B=\left[\begin{array}{cc}{B}_{11}& {B}_{12}\\ {B}_{21}& {B}_{22}\end{array}\right]$ (2)

$M=\left[\begin{array}{ccc}{A}_{11}& {A}_{12}& O\\ {A}_{21}& {A}_{22}+{B}_{11}& {B}_{12}\\ O& {B}_{21}& {B}_{22}\end{array}\right]\in {R}^{n×n}$ (3)

$A=\left({a}_{ij}\right)\in {R}^{{n}_{1}×{n}_{1}},\text{\hspace{0.17em}}B=\left({b}_{ij}\right)\in {R}^{{n}_{2}×{n}_{2}}$$M=\left({m}_{ij}\right)\in {R}^{n×n}$ 按定义2.2中的(1)式分别分裂为：

$A={A}^{z}+{A}^{r}$$B={B}^{z}+{B}^{r}$$M=A{\oplus }_{k}B={M}^{z}+{M}^{r}$

${A}^{z},{A}^{r},{B}^{z},{B}^{r}$ 按(2)式分块为：

${A}^{z}=\left[\begin{array}{cc}{A}_{11}^{z}& {A}_{12}^{z}\\ {A}_{21}^{z}& {A}_{22}^{z}\end{array}\right]$${A}^{r}=\left[\begin{array}{cc}{A}_{11}^{r}& {A}_{12}^{r}\\ {A}_{21}^{r}& {A}_{22}^{r}\end{array}\right]$${B}^{z}=\left[\begin{array}{cc}{B}_{11}^{z}& {B}_{12}^{z}\\ {B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]$${B}^{r}=\left[\begin{array}{cc}{B}_{11}^{r}& {B}_{12}^{r}\\ {B}_{21}^{r}& {B}_{22}^{r}\end{array}\right]$

$\stackrel{¯}{M}=\left[\begin{array}{ccc}{A}_{11}^{z}& {A}_{12}^{z}& -{A}_{13}^{r}\\ {A}_{21}^{z}-{B}_{13}^{r}& {A}_{22}^{z}+{B}_{11}^{z}& {B}_{12}^{z}-{A}_{23}^{r}\\ -{B}_{23}^{r}& {B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]\in {R}^{n×n}$ (4)

${A}_{11}^{z}\in {R}^{\left({n}_{1}-k\right)×\left({n}_{1}-k\right)}$${A}_{12}^{z}\in {R}^{\left({n}_{1}-k\right)×k}$${A}_{21}^{z}\in {R}^{k×\left({n}_{1}-k\right)}$${A}_{22}^{z}\in {R}^{k×k}$

${B}_{11}^{z}\in {R}^{k×k}$${B}_{12}^{z}\in {R}^{k×\left(n-{n}_{1}\right)}$${B}_{21}^{z}\in {R}^{\left(n-{n}_{1}\right)×k}$${B}_{22}^{z}\in {R}^{\left(n-{n}_{1}\right)×\left(n-{n}_{1}\right)}$

${A}_{11}^{r}\in {R}^{\left({n}_{1}-k\right)×\left({n}_{1}-k\right)}$${A}_{12}^{r}\in {R}^{\left({n}_{1}-k\right)×k}$${A}_{13}^{r}\in {R}^{\left({n}_{1}-k\right)\left(n-{n}_{1}\right)}$ 的第i行为 $\left({\beta }_{i}^{A},{\beta }_{i}^{A},\cdots ,{\beta }_{i}^{A}\right)$

${A}_{21}^{r}\in {R}^{k×\left({n}_{1}-k\right)}$${A}_{22}^{r}\in {R}^{k×k}$${A}_{23}^{r}\in {R}^{k×\left(n-{n}_{1}\right)}$ 的第i行为 $\left({\beta }_{i}^{A},{\beta }_{i}^{A},\cdots ,{\beta }_{i}^{A}\right)$

${B}_{11}^{r}\in {R}^{k×k}$${B}_{12}^{r}\in {R}^{k×\left(n-{n}_{1}\right)}$${B}_{13}^{r}\in {R}^{k×\left({n}_{1}-k\right)}$ 的第i行为 $\left({\beta }_{i}^{B},{\beta }_{i}^{B},\cdots ,{\beta }_{i}^{B}\right)$

${B}_{21}^{r}\in {R}^{\left(n-{n}_{1}\right)×k}$${B}_{22}^{r}\in {R}^{\left(n-{n}_{1}\right)×\left(n-{n}_{1}\right)}$${B}_{23}^{r}\in {R}^{\left(n-{n}_{1}\right)×\left({n}_{1}-k\right)}$ 的第i行为 $\left({\beta }_{i}^{B},{\beta }_{i}^{B},\cdots ,{\beta }_{i}^{B}\right)$

${A}^{z},{B}^{z}$ 为非奇异矩阵时，将 ${\left({A}^{z}\right)}^{-1},{\left({B}^{z}\right)}^{-1}$ 按(2)分块为：

${\left({A}^{z}\right)}^{-1}=\left[\begin{array}{cc}\stackrel{^}{{A}_{11}^{z}}& \stackrel{^}{{A}_{12}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}& \stackrel{^}{{A}_{22}^{z}}\end{array}\right]$${\left({B}^{z}\right)}^{-1}=\left[\begin{array}{cc}\stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]$ (*)

1) 当 $A$ 为非奇异M-矩阵时，其主对角元为正。

2) 当 $A$ 为非奇异M-矩阵， $B=\left[{b}_{ij}\right]$ 为Z-矩阵且 $B\ge A$ 时， $B$ 为非奇异M-矩阵。

3) $A$ 为非奇异M-矩阵的充要条件为 $A$ 的每一个主子矩阵为非奇异M-矩阵。

1) 若 $D$ 非奇异且 ${D}^{-1}\ge 0$$-E\ge 0$$-F\ge 0$ ，则 ${A}^{-1}\ge 0$ 当且仅当 ${\left(A/D\right)}^{-1}\ge 0$

2) 若 ${G}^{-1}\ge 0$$-E\ge 0$$-F\ge 0$ ，则 ${A}^{-1}\ge 0$ 当且仅当 ${\left(A/G\right)}^{-1}\ge 0$

3. MB-矩阵的k-子直和

$\mathrm{det}\stackrel{^}{H}=\mathrm{det}\left\{\left(\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}\right)+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\right\}\ne 0$

${B}_{23}^{r}=O$ ，则 $\stackrel{¯}{M}$ 为非奇异的Z-矩阵。

${\left({A}^{z}\right)}^{-1}\left[\begin{array}{cc}{I}_{{n}_{1}-k}& O\\ {B}_{13}^{r}& {I}_{k}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}\end{array}\right]$

$\mathrm{det}\left[\begin{array}{cc}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}\end{array}\right]\ne 0$

${A}_{11}^{z}\stackrel{^}{{A}_{11}^{z}}+{A}_{12}^{z}\stackrel{^}{{A}_{12}^{z}}={I}_{{n}_{1}-k}={I}_{n-{n}_{2}}$${A}_{11}^{z}\stackrel{^}{{A}_{12}^{z}}+{A}_{12}^{z}\stackrel{^}{{A}_{22}^{z}}=0$

${A}_{21}^{z}\stackrel{^}{{A}_{11}^{z}}+{A}_{22}^{z}\stackrel{^}{{A}_{21}^{z}}=0$${A}_{21}^{z}\stackrel{^}{{A}_{12}^{z}}+{A}_{22}^{z}\stackrel{^}{{A}_{22}^{z}}={I}_{k}$

${\left({A}^{z}\right)}^{-1}{A}^{z}={I}_{{n}_{1}}$ ，得

$\stackrel{^}{{A}_{11}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{12}^{z}={I}_{n-{n}_{2}}$$\stackrel{^}{{A}_{11}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{22}^{z}=0$

$\stackrel{^}{{A}_{21}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{22}^{z}}{A}_{21}^{z}=0$$\stackrel{^}{{A}_{21}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}{A}_{22}^{z}={I}_{k}$

${B}^{z}{\left({B}^{z}\right)}^{-1}={I}_{{n}_{2}}$ ，得

${B}_{11}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{21}^{z}}={I}_{k}$${B}_{11}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{22}^{z}}=0$

${B}_{21}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{21}^{z}}=0$${B}_{21}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{22}^{z}}={I}_{n-{n}_{1}}$

${\left({B}^{z}\right)}^{-1}{B}^{z}={I}_{{n}_{2}}$ ，得

$\stackrel{^}{{B}_{11}^{z}}{B}_{11}^{z}+\stackrel{^}{{B}_{12}^{z}}{B}_{21}^{z}={I}_{k}$$\stackrel{^}{{B}_{11}^{z}}{B}_{12}^{z}+\stackrel{^}{{B}_{12}^{z}}{B}_{22}^{z}=0$

$\stackrel{^}{{B}_{21}^{z}}{B}_{11}^{z}+\stackrel{^}{{B}_{22}^{z}}{B}_{21}^{z}=0$$\stackrel{^}{{B}_{21}^{z}}{B}_{12}^{z}+\stackrel{^}{{B}_{22}^{z}}{B}_{22}^{z}={I}_{n-{n}_{1}}$

$\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}& 0\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}& 0\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]\stackrel{¯}{M}\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ 0& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]$

$=\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}& 0\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}& 0\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]\left[\begin{array}{ccc}{A}_{11}^{z}& {A}_{12}^{z}& -{A}_{13}^{r}\\ {A}_{21}^{z}-{B}_{13}^{r}& {A}_{22}^{z}+{B}_{11}^{z}& {B}_{12}^{z}-{A}_{23}^{r}\\ -{B}_{23}^{r}& {B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ 0& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]$ $=\left[\begin{array}{cc}\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}\left({A}_{21}^{z}-{B}_{13}^{r}\right)& \left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\\ \left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{11}^{z}+\stackrel{^}{{A}_{22}^{z}}\left({A}_{21}^{z}-{B}_{13}^{r}\right)& \left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\\ -{B}_{23}^{r}& {B}_{21}^{z}\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}-\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\\ -\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\\ {B}_{22}^{z}\end{array}\right]\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ 0& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]$

$=\left[\begin{array}{c}\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}\left({A}_{21}^{z}-{B}_{13}^{r}\right)\\ \left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{11}^{z}+\stackrel{^}{{A}_{22}^{z}}\left({A}_{21}^{z}-{B}_{13}^{r}\right)\\ -{B}_{23}^{r}\end{array}\text{\hspace{0.17em}}$ $\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\left(\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\right)\stackrel{^}{{B}_{11}^{z}}+\left(-\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\right)\stackrel{^}{{B}_{21}^{z}}\\ \left(\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\right)\stackrel{^}{{B}_{11}^{z}}+\left(-\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\right)\stackrel{^}{{B}_{21}^{z}}\\ {B}_{21}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{21}^{z}}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\left(\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\right)\stackrel{^}{{B}_{12}^{z}}+\left(-\left(\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\right)\stackrel{^}{{B}_{22}^{z}}\\ \left(\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}\left({A}_{22}^{z}+{B}_{11}^{z}\right)\right)\stackrel{^}{{B}_{12}^{z}}+\left(-\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right){A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}\left({B}_{12}^{z}-{A}_{23}^{r}\right)\right)\stackrel{^}{{B}_{22}^{z}}\\ {B}_{21}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{22}^{z}}\end{array}\right]$

$\begin{array}{l}=\left[\begin{array}{c}\stackrel{^}{{A}_{11}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{21}^{z}-\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\\ \stackrel{^}{{A}_{21}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}{A}_{11}^{z}+{A}_{22}^{z}{A}_{21}^{z}-\stackrel{^}{{A}_{22}^{z}}\stackrel{^}{{B}_{13}^{r}}\\ -{B}_{23}^{z}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\stackrel{^}{{A}_{11}^{z}}{A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}{A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{A}_{22}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{11}^{z}\stackrel{^}{{B}_{11}^{z}}-\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}-\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{12}^{z}\stackrel{^}{{B}_{21}^{z}}-\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\stackrel{^}{{B}_{21}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}{A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}{A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}{A}_{22}^{z}\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{11}^{z}\stackrel{^}{{B}_{11}^{z}}-\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}-\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{12}^{z}\stackrel{^}{{B}_{21}^{z}}-\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\stackrel{^}{{B}_{21}^{z}}\\ {B}_{21}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{21}^{z}}\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\stackrel{^}{{A}_{11}^{z}}{A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}{A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{A}_{22}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{11}^{z}\stackrel{^}{{B}_{12}^{z}}-\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}-\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{12}^{z}\stackrel{^}{{B}_{22}^{z}}-\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\stackrel{^}{{B}_{22}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}{A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}{A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{A}_{22}^{z}\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{11}^{z}\stackrel{^}{{B}_{12}^{z}}-\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}-\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{12}^{z}\stackrel{^}{{B}_{22}^{z}}-\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\stackrel{^}{{B}_{22}^{z}}\\ {B}_{21}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{22}^{z}\stackrel{^}{{B}_{22}^{z}}\end{array}\right]$

$=\left[\begin{array}{cc}\stackrel{^}{{A}_{11}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{21}^{z}& \left(\stackrel{^}{{A}_{11}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{22}^{z}\right)\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)+\stackrel{^}{{A}_{12}^{z}}\left({B}_{11}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\\ \stackrel{^}{{A}_{21}^{z}}{A}_{11}^{z}+\stackrel{^}{{A}_{22}^{z}}{A}_{21}^{z}& \left(\stackrel{^}{{A}_{21}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}{A}_{22}^{z}\right)\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)+\stackrel{^}{{A}_{22}^{z}}\left({B}_{11}^{z}\stackrel{^}{{B}_{11}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\\ -{B}_{23}^{z}& 0\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\left(\stackrel{^}{{A}_{11}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{12}^{z}}{A}_{22}^{z}\right)\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)+\stackrel{^}{{A}_{12}^{z}}\left({B}_{11}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ \left(\stackrel{^}{{A}_{21}^{z}}{A}_{12}^{z}+\stackrel{^}{{A}_{22}^{z}}{A}_{22}^{z}\right)\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)+\stackrel{^}{{A}_{22}^{z}}\left({B}_{11}^{z}\stackrel{^}{{B}_{12}^{z}}+{B}_{12}^{z}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ {I}_{n-{n}_{1}}\end{array}\right]$ $=\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& \stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ 0& \stackrel{^}{H}& \stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ -{B}_{23}^{r}& 0& {I}_{n-{n}_{1}}\end{array}\right]$

${\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& F& Y\\ 0& \stackrel{^}{H}& Q\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]}^{-1}\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& C& D\\ O& {\stackrel{^}{H}}^{-1}& E\\ O& O& {I}_{n-{n}_{1}}\end{array}\right]=\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& {I}_{k}& 0\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]$

$C=-\left[\stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\right]×{\stackrel{^}{H}}^{-1}$

$\begin{array}{c}D=-\left[\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\stackrel{^}{H}}^{-1}×\left[\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\right]\end{array}$

$E=-{\stackrel{^}{H}}^{-1}×\left[\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\right]$

$F=\stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}$

$Y=\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}$

$Q=\stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}$

$\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}& 0\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}& 0\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]\stackrel{¯}{M}\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ 0& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]$

$=\left[\begin{array}{cc}{I}_{n-{n}_{2}}& \stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\\ 0& \stackrel{^}{H}\\ -{B}_{23}^{r}& 0\end{array}$ $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ \stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ {I}_{n-{n}_{1}}\end{array}\right]$

${\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& 0& 0\\ 0& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ 0& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]}^{-1}{\left(\stackrel{¯}{M}\right)}^{-1}{\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}& 0\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}& 0\\ 0& 0& {I}_{n-{n}_{1}}\end{array}\right]}^{-1}$

$\begin{array}{l}=\left\{\left[\begin{array}{cc}{I}_{n-{n}_{2}}& \stackrel{^}{{A}_{12}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\\ 0& \stackrel{^}{H}\\ -{B}_{23}^{r}& 0\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\begin{array}{c}\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ \stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ {I}_{n-{n}_{1}}\end{array}\right]\right\}}^{-1}$

${\begin{array}{c}\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{11}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{12}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ \stackrel{^}{{B}_{12}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{12}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{22}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{22}^{z}}\\ {I}_{n-{n}_{1}}\end{array}\right]\right\}}^{-1}$

${\left(\stackrel{¯}{M}\right)}^{-1}=\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& O& O\\ O& \stackrel{^}{{B}_{11}^{z}}& \stackrel{^}{{B}_{12}^{z}}\\ O& \stackrel{^}{{B}_{21}^{z}}& \stackrel{^}{{B}_{22}^{z}}\end{array}\right]\left[\begin{array}{ccc}{I}_{n-{n}_{2}}& C& D\\ O& {\stackrel{^}{H}}^{-1}& E\\ O& O& {I}_{n-{n}_{1}}\end{array}\right]\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{12}^{z}}& O\\ \stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}& \stackrel{^}{{A}_{22}^{z}}& O\\ O& O& {I}_{n-{n}_{1}}\end{array}\right]$

${\left(\stackrel{¯}{M}\right)}^{-1}=\left[\begin{array}{ccc}\stackrel{^}{{A}_{11}^{z}}+\stackrel{^}{{A}_{12}^{z}}{B}_{13}^{r}+C\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right)& \stackrel{^}{{A}_{12}^{z}}+C\stackrel{^}{{A}_{22}^{z}}& D\\ \stackrel{^}{{B}_{11}^{z}}\stackrel{^}{H}\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right)& \stackrel{^}{{B}_{11}^{z}}{\stackrel{^}{H}}^{-1}\stackrel{^}{{A}_{22}^{z}}& \stackrel{^}{{B}_{11}^{z}}E+\stackrel{^}{{B}_{12}^{z}}\\ \stackrel{^}{{B}_{21}^{z}}{\stackrel{^}{H}}^{-1}\left(\stackrel{^}{{A}_{21}^{z}}+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\right)& \stackrel{^}{{B}_{21}^{z}}{\stackrel{^}{H}}^{-1}\stackrel{^}{{A}_{22}^{z}}& \stackrel{^}{{B}_{21}^{z}}E+\stackrel{^}{{B}_{22}^{z}}\end{array}\right]$ (5)

$\mathrm{det}\stackrel{^}{H}=\mathrm{det}\left\{\left(\stackrel{^}{{B}_{11}^{z}}+\stackrel{^}{{A}_{22}^{z}}\right)+\stackrel{^}{{A}_{22}^{z}}{B}_{13}^{r}\left({A}_{12}^{z}\stackrel{^}{{B}_{11}^{z}}-{A}_{13}^{r}\stackrel{^}{{B}_{21}^{z}}\right)-\left(\stackrel{^}{{A}_{21}^{z}}{A}_{13}^{r}+\stackrel{^}{{A}_{22}^{z}}{A}_{23}^{r}\right)\stackrel{^}{{B}_{21}^{z}}\right\}\ne 0$

${B}_{23}^{r}=O$ ，则当(5)式中的每一个分块为非负矩阵时， $M=A{\oplus }_{k}B$ 为MB-矩阵。

${\left(\stackrel{˜}{D}\right)}^{-1}\ge 0$${\left(\stackrel{˜}{G}-\stackrel{˜}{F}{\left(\stackrel{˜}{D}\right)}^{-1}\stackrel{˜}{E}\right)}^{-1}\ge 0$

$M=A{\oplus }_{k}B$ 为MB-矩阵，其中

$\stackrel{˜}{D}={A}_{11}^{z}-{A}_{13}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}$$\stackrel{˜}{E}={A}_{12}^{z}+{A}_{13}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}\le 0$

$\stackrel{˜}{F}={A}_{21}^{z}-{B}_{13}^{r}+{B}_{12}^{z}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}-{A}_{23}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}\le 0$$\stackrel{˜}{G}={A}_{22}^{z}+{B}_{11}^{z}-{B}_{12}^{z}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}+{A}_{23}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}$

$\stackrel{¯}{M}/T=\left[\begin{array}{cc}{A}_{11}^{z}-{A}_{13}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}& {A}_{12}^{z}+{A}_{13}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}\\ {A}_{21}^{z}-{B}_{13}^{r}+{B}_{12}^{z}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}-{A}_{23}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{23}^{r}& {A}_{22}^{z}+{B}_{11}^{z}-{B}_{12}^{z}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}+{A}_{23}^{r}{\left({B}_{22}^{z}\right)}^{-1}{B}_{21}^{z}\end{array}\right]$

$\stackrel{¯}{M}/T=\left[\begin{array}{cc}\stackrel{˜}{D}& \stackrel{˜}{E}\\ \stackrel{˜}{F}& \stackrel{˜}{G}\end{array}\right]$ ，则当 ${\left(\stackrel{˜}{D}\right)}^{-1}\ge 0$ 时，由引理2.1得 ${\left(\stackrel{¯}{M}/T\right)}^{-1}\ge 0$ 当且仅当

${\left(\left(\stackrel{¯}{M}/T\right)/\stackrel{˜}{D}\right)}^{-1}={\left(\stackrel{˜}{G}-\stackrel{˜}{F}{\left(\stackrel{˜}{D}\right)}^{-1}\stackrel{˜}{E}\right)}^{-1}\ge 0$

$C=\left[\begin{array}{ccc}{A}_{11}^{z}& {A}_{12}^{z}& -{A}_{13}^{r}\\ {A}_{21}^{z}-{B}_{13}^{r}& {A}_{22}^{z}& {B}_{12}^{z}-{A}_{23}^{r}\\ -{B}_{23}^{r}& {B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]$

$T=\left[\begin{array}{ccc}{A}_{11}^{z}& 2{A}_{12}^{z}& -{A}_{13}^{r}\\ {A}_{21}^{z}-{B}_{13}^{r}& 2{A}_{22}^{z}& {B}_{12}^{z}-{A}_{23}^{r}\\ -{B}_{23}^{r}& 2{B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]$

$T=Cdiag\left(I,2I,I\right)$${T}^{-1}=diag\left(I,\left(1/2\right)I,I\right){C}^{-1}\ge O$$T$ 是一个非奇异的M-矩阵。此时

$\stackrel{¯}{M}=\left[\begin{array}{ccc}{A}_{11}^{z}& {A}_{12}^{z}& -{A}_{13}^{r}\\ {A}_{21}^{z}-{B}_{13}^{r}& 2{A}_{22}^{z}& {B}_{12}^{z}-{A}_{23}^{r}\\ -{B}_{23}^{r}& {B}_{21}^{z}& {B}_{22}^{z}\end{array}\right]$

$\stackrel{¯}{M}\ge T$ ，于是 $\stackrel{¯}{M}$ 为非奇异的M-矩阵。由 ${M}^{z}\ge \stackrel{¯}{M}$${M}^{z}$ 也为非奇异的M-矩阵。

$A=\left[\begin{array}{cccc}4& .& 1& 1\\ .& .& .& .\\ -1& .& 11& -3\\ -1& .& -13& 8\end{array}\right]={A}^{z}+{A}^{r}=\left[\begin{array}{ccc}3& 0& 0\\ -1& 11& -3\\ -1& -13& 8\end{array}\right]+\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$B=\left[\begin{array}{cccc}11& -2& .& -1\\ -13& 8& .& -3\\ .& .& .& .\\ 1& 1& .& 3\end{array}\right]\left[\begin{array}{ccc}11& -3& -1\\ -13& 8& -3\\ 1& 1& 3\end{array}\right]={B}^{z}+{B}^{r}=\left[\begin{array}{ccc}11& -3& -1\\ -13& 8& -3\\ 0& 0& 2\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 1& 1\end{array}\right]$

$C=\left[\begin{array}{cccc}3& 0& 0& -1\\ -1& 11& -3& -1\\ -1& -13& 8& -3\\ -1& 0& 0& 2\end{array}\right]$

$M=A{\oplus }_{2}B=\left[\begin{array}{cccc}4& 1& 1& 0\\ -1& 22& -6& -1\\ -1& -26& 16& -3\\ 0& 1& 1& 3\end{array}\right]$

Conditions That Subdirect Sums of MB-Matrices Is Still MB-Matrices[J]. 理论数学, 2017, 07(06): 422-430. http://dx.doi.org/10.12677/PM.2017.76055

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