# The maximal and minimal ranks of matrix expression with applications

## Abstract

We give in this article the maximal and minimal ranks of the matrix expression A-B1V1C1-B2V2C2-B3V3C3-B4V4C4 with respect to V1, V2, V3, and V4. As applications, we derive the extremal ranks of the generalized Schur complement A - BM(1)C - DN(1)G and the partial matrix (A BM(1)C DN(1)G) with respect to the generalized inverse M(1) ε M{1} and N(1) N{1}.

AMS classifications: 15A03; 15A09; 15A24.

## 1 Introduction

Let Cm × n be the set of all m × n complex matrices with complex entries. I n denotes the identity matrix of order n and O m × n denotes the m × n matrix of all zero entries (if no confusion occurs, we will omit the subscript). For a given a matrix A Cm× n, the symbols A* and r(A) will stand for the conjugate transpose and the rank of the matrix A, respectively. We recall that a generalized inverse X Cn× mof A Cm × nis a matrix which satisfies some of the following four Penrose equations :

$\left(1\right)AXA=A,\phantom{\rule{1em}{0ex}}\left(2\right)\phantom{\rule{0.3em}{0ex}}XAX=X,\phantom{\rule{1em}{0ex}}\left(3\right)\phantom{\rule{0.3em}{0ex}}{\left(AX\right)}^{*}=AX,\phantom{\rule{1em}{0ex}}\left(4\right)\phantom{\rule{0.3em}{0ex}}{\left(XA\right)}^{*}=XA.$

For a subset {i,j,...,k} of the set {1,2,3,4}, the set of n × m matrices satisfying the equations (i), (j), ...,(k) from among the above four Penrose Equations (1)-(4) is denoted by A{i,j,...,k}. A matrix X from A{i,j,...,k} is called an {i,j,...,k}-inverse of A and is denoted by A(i,j,...,k). In particular, an n × m matrix X of the set A{1} is called a g-inverse of A and denoted by A(1). The unique {1, 2, 3, 4}-inverse of A is denoted by A, which is called the Moore-Penrose inverse of A. Throughout this article, the abbreviated symbols E A and F A stand for the two projectors E A = I - AA and F A = I − AA of A, respectively. We refer the reader to [2, 3] for basic results on the generalized inverses.

Given a matrix with some variant entries in it (often called partial matrix) or a matrix expression with some variant matrices in it, the rank of the partial matrix or matrix expression will vary with respect to the variant entries or variant matrices. Because the rank of matrix is an integer between 0 and the minimal of row and column numbers of the matrix, maximal and minimal ranks of partial matrix or matrix expressions must exist with respect to their variant entries or variant matrices. Many problems in matrix theory and applications are closely related to maximal and minimal possible ranks of matrix expressions with variant entries. For example, a matrix equation AXB = C is consistent if and only if the minimal rank of C-AXB with respect to X is zero, see ; there is matrix X such that the partial matrix AXB of order n is nonsingular if and only if the maximal rank of AXB with respect to X is n, see .

The maximal and minimal ranks of matrix expressions or partial matrix are two basic concepts in matrix theory for describing the dimension of the row or column vector space of matrix expressions or partial matrix, both of which are well understood and are easy to compute by the well-known elementary or congruent matrix operations, see [5, 7, 8, 1016]. These two quantities play an essential role in characterizing algebraic properties of matrices expressions or partial matrices. In fact, maximal and minimal ranks of matrix expressions or partial matrices have been the main objects of study in matrix theory and applications. Some previous systematical researches on maximal and minimal ranks of matrix expressions or partial matrices and their applications can be found in . In recent years, the present author reconsidered the maximal and minimal ranks of matrix expressions or partial matrices by using some tricky operations on block matrices and generalized inverses of matrices, and obtained many new formulas for maximal and minimal ranks of matrix expressions or partial matrices and their applications, see [4, 6, 9, 2128].

In this article, given matrices $A\in {C}^{m×n},\phantom{\rule{2.77695pt}{0ex}}{B}_{i}\in {C}^{m×{p}_{i}},\phantom{\rule{2.77695pt}{0ex}}{C}_{i}\in {C}^{{q}_{i}×n},\phantom{\rule{2.77695pt}{0ex}}i=1,2,3,4$, we will present the maximal and minimal ranks of the matrix expression A-B1V1C1 - B2V2C2 - B3V3C3 - B4V4C4 with respect to V1, V2, V3, and V4. As applications, the maximal and minimal ranks of the generalized Schur complement A - BM(1)C - DN(1)G and the partial matrix (A BM(1)C DN(1)G) with respect to the generalized inverse M(1) M{1} and N(1) N{1} are also considered. The results in this article extend the earlier studies by various authors, see, e.g., [46, 11, 16, 18, 21, 25, 26].

We first introduce some well-known results which will be used in this article.

Lemma 1.1 [5, 8, 25]. Let

$M=\left(\begin{array}{ccc}\hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill X\hfill \\ \hfill {A}_{21}\hfill & \hfill {A}_{22}\hfill & \hfill {A}_{23}\hfill \\ \hfill Y\hfill & \hfill {A}_{32}\hfill & \hfill {A}_{33}\hfill \end{array}\right)$

where ${A}_{ij}\in {C}^{{m}_{i}×{n}_{j}}\left(1\le i,\phantom{\rule{2.77695pt}{0ex}}j\le 3\right)$ are given, $X\in {C}^{{m}_{1}×{n}_{3}}$ and $Y\in {C}^{{m}_{3}×{n}_{1}}$ are two variant matrices. Then

(1)
(2)

Lemma 1.2 . Let A Cm × n. Then the expression of {1}-inverses of A can be written as

$\begin{array}{c}\hfill {A}^{\left(1\right)}={A}^{†}+\left({I}_{n}-{A}^{†}A\right)W+Z\left({I}_{m}-A{A}^{†}\right),\hfill \end{array}$
(3)

where W Cn × mand Z Cn× mare arbitrary.

Lemma 1.3 . Let A Cm × n, B Cm × k, and C Cl × n. Then

$\begin{array}{c}\left(1\right).r\left(AB\right)=r\left(A\right)+r\left({E}_{A}B\right)=r\left({E}_{B}A\right)+r\left(B\right),\\ \left(2\right).r\left(\begin{array}{c}\hfill A\hfill \\ \hfill C\hfill \end{array}\right)=r\left(A\right)+r\left(C{F}_{A}\right)=r\left(A{F}_{C}\right)+r\left(C\right),\end{array}$

where E A = I m - AA and F A = I n - AA.

## 2 The maximal and minimal ranks of A - B1V1C1- B2V2C2- B3V3C3- B4V4C4

In this section, we will present the maximal and minimal ranks of the linear matrix expression

$P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)=A-{B}_{1}{V}_{1}{C}_{1}-{B}_{2}{V}_{2}{C}_{2}-{B}_{3}{V}_{3}{C}_{3}-{B}_{4}{V}_{4}{C}_{4},$
(4)

where $A\in {C}^{m×n},\phantom{\rule{2.77695pt}{0ex}}{B}_{i}\in {C}^{m×{p}_{i}},\phantom{\rule{2.77695pt}{0ex}}{C}_{i}\in {C}^{{q}_{i}×n},\phantom{\rule{2.77695pt}{0ex}}i=1,2,3,4$, are given matrices, with respect to four variant matrices ${V}_{i}\in {C}^{{p}_{i}×{q}_{i}},\phantom{\rule{2.77695pt}{0ex}}i=1,2,3,4$. Applying the formula (1) in Lemma 1.1 to the linear matrix expression in (4) and simplifying, we obtain the following result.

Theorem 2.1 Let P(V1, V2, V3, V4) be given as (4). Then

$\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{max}}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=\text{min}\left\{{T}_{1},{T}_{2},{T}_{3},{T}_{4}\right\},\hfill \end{array}$
(5)

where

$\begin{array}{c}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=r\left(\begin{array}{ccccccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{4}}\hfill & \hfill -{V}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{q}_{4}}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{2}}\hfill & \hfill -{V}_{2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {B}_{3}\hfill & \hfill O\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill O\hfill & \hfill {B}_{4}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill -{V}_{1}\hfill & \hfill {I}_{{P}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill {I}_{{q}_{3}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{3}\hfill & \hfill {I}_{{P}_{3}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)-\sum _{i=1}^{4}{p}_{i}-\sum _{i=1}^{4}{q}_{i},\\ =r\left(T\right)-\sum _{i=1}^{4}pi-\sum _{i=1}^{4}qi,\end{array}$

Proof. It is easy to verify by block Gaussian elimination that the rank of P(V1,V2,V3, V4) in (4) can be expressed as

$\begin{array}{c}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=r\left(\begin{array}{ccccccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{P4}\hfill & \hfill -{V}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{q4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{P2}\hfill & \hfill -{V}_{2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill {I}_{q2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {B}_{3}\hfill & \hfill O\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill O\hfill & \hfill {B}_{4}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {I}_{q1}\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill -{V}_{1}\hfill & \hfill {I}_{P1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill {I}_{q3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{3}\hfill & \hfill {I}_{P3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)-\sum _{i=1}^{4}pi-\sum _{i=1}^{4}qi,\\ =r\left(T\right)-\sum _{i=1}^{4}pi-\sum _{i=1}^{4}qi,\end{array}$

where $A\in {C}^{m×n},\phantom{\rule{2.77695pt}{0ex}}{B}_{i}\in {C}^{m×{p}_{i}},\phantom{\rule{2.77695pt}{0ex}}{C}_{i}\in {C}^{{q}_{i}×n},\phantom{\rule{2.77695pt}{0ex}}{V}_{i}\in {C}^{{p}_{i}×{q}_{i}},\phantom{\rule{2.77695pt}{0ex}}i=1,2,3,4$ and ${I}_{{p}_{i}},{I}_{{q}_{i}},i=1,2,3,4$, are denotes the identity matrix of order p i and q i , respectively.

$\begin{array}{c}\hfill T=\left(\begin{array}{ccc}\hfill O\hfill & \hfill {E}_{2}\hfill & \hfill -{V}_{4}\hfill \\ \hfill {E}_{1}\hfill & \hfill S\hfill & \hfill {E}_{3}\hfill \\ \hfill -{V}_{3}\hfill & \hfill {E}_{4}\hfill & \hfill O\hfill \end{array}\right),S=\left(\begin{array}{ccccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{p}_{2}}\hfill & \hfill -{V}_{2}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill O\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill O\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill -{V}_{1}\hfill & \hfill {I}_{{p}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\hfill \end{array}$

and

$\begin{array}{c}{E}_{1}={\left(O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}{I}_{{q}_{3}}\right)}^{*},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{E}_{\text{2}}=\left(O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}{I}_{{p}_{4}}\right),\\ {E}_{3}={\left({I}_{{q}_{4}}\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\right)}^{*},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{E}_{4}=\left({I}_{{p}_{3}}\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\phantom{\rule{0.3em}{0ex}}O\right).\end{array}$

According to this result, we have

$\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{max}}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=\underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{max}}r\left(T\right)-\sum _{i=1}^{4}{p}_{i}-\sum _{i=1}^{4}{q}_{i}.\hfill \end{array}$
(6)

Then applying the formula (1) in Lemma 1.1 to matrix T, we have

where

$\begin{array}{c}\begin{array}{c}\hfill {S}_{1}=\left(\begin{array}{cccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{2}}\hfill & \hfill -{V}_{2}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill \\ \hfill O\hfill & \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill O\hfill \\ \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{1}\hfill & \hfill O\hfill & \hfill {I}_{{p}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),{S}_{2}=\left(\begin{array}{cccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{2}}\hfill & \hfill O\hfill & \hfill -{V}_{2}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill \\ \hfill O\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill & \hfill O\hfill \\ \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{1}\hfill & \hfill {I}_{{p}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\hfill \end{array}\\ {S}_{3}\left(\begin{array}{ccccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{2}}\hfill & \hfill O\hfill & \hfill -{V}_{2}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill \\ \hfill O\hfill & \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill & \hfill O\hfill \\ \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{1}\hfill & \hfill O\hfill & \hfill {I}_{{p}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),{S}_{4}=\left(\begin{array}{ccccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill & \hfill {I}_{{P}_{2}}\hfill & \hfill -{V}_{2}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill & \hfill {I}_{{q}_{2}}\hfill \\ \hfill O\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill O\hfill \\ \hfill {I}_{{q}_{1}}\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill -{V}_{1}\hfill & \hfill {I}_{{p}_{1}}\hfill & \hfill O\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\end{array}$

Again applying the formula (1) in Lemma 1.1, we get

(7)

and

(8)
(9)
$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{max}}r\left({S}_{3}\right)=\text{min}\left\{{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right),\right\\hfill \end{array}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{cccc}\hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left({B}_{3}{B}_{1}A{B}_{2}{B}_{4}\right),\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right),\end{array}$
(10)
$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{max}}r\left({S}_{4}\right)=\text{min}\left\{{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right),{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{cc}\hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),\right\\hfill \end{array}\\ {p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),{p}_{1}+{q}_{2}+{q}_{1}+{p}_{2}+r\left(\begin{array}{c}\hfill {C}_{4}\hfill \\ \hfill {C}_{2}\hfill \\ \hfill A\hfill \\ \hfill {C}_{1}\hfill \\ \hfill {C}_{3}\hfill \end{array}\right).\end{array}$
(11)

Substituting (8)-(11) into (7) and (6) yield (5).

Recall a simple fact that a matrix equation AXB = C is consistent for every variant matrices X, if and only if the maximal rank of C - AXB with respect to X is zero. Thus, by Theorem 2.1 we can immediately obtain the following result.

Corollary 2.2 Let P(V1,V2,V3, V4) be given as (4). Then the matrix equation A = B1V1C1 + B2V2C2 + B3V3C3 + B4V4C4 holds for any V1, V2, V3, and V4 if and only if T1 = O or T2 = O or T3 = O or T4 = O.

Because the right side of (5) are just composed by ranks of block matrices, they can be easily simplified by block Gaussian elimination when the given matrices in (4) satisfy some restrictions.

Theorem 2.3 Let P(V1, V2, V3, V4) be given as (4) and let R(B1) R(B2), R(B3) R(B4), $R\left({B}_{1}\right)\subseteq R\left({B}_{2}\right),R\left({B}_{3}\right)\subseteq R\left({B}_{4}\right),R\left({C}_{2}^{\ast }\right)\subseteq R\left({C}_{1}^{\ast }\right),R\left({C}_{4}^{\ast }\right)\subseteq R\left({C}_{3}^{\ast }\right)$ Then

$\underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{max}}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=\text{min}\left\{{\tau }_{1},{\tau }_{2},{\tau }_{3}\right\},$
(12)

where

$\begin{array}{c}{\tau }_{1}=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \end{array}\right)\right\},\\ \phantom{\rule{2.77695pt}{0ex}}{\tau }_{2}=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right),r\left(A{B}_{2}{B}_{4}\right),r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)\right\},\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\tau }_{3}=\text{min}\left\{r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{c}\hfill A\hfill \\ \hfill {C}_{1}\hfill \\ \hfill {C}_{3}\hfill \end{array}\right)\right\}.\end{array}$

Proof. In fact, we can write B1 = B2X, B3 = B4Y, C2 = ZC1, and C4 = WC3 under the hypotheses of Theorem 2.3. In this case, we have

$\begin{array}{c}\begin{array}{c}\hfill r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \end{array}\right),\hfill \end{array}\\ r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill Z{C}_{1}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{2}X\hfill & \hfill A\hfill \end{array}\right)\le r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)\end{array}$
(13)

and

$\begin{array}{c}\begin{array}{c}\hfill r\left(\begin{array}{cccc}\hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),\hfill \end{array}\\ r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill Z{C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}X\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)\le r\left(\begin{array}{ccc}\hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\end{array}$
(14)

and

$\begin{array}{c}\begin{array}{c}\hfill r\left({B}_{3}{B}_{1}A{B}_{2}{B}_{4}\right)=r\left(A{B}_{2}{B}_{4}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right),\hfill \end{array}\\ r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)=r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill Z{C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{2}X\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\le r\left(\begin{array}{cccc}\hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),\\ r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\end{array}$
(15)

and

$\begin{array}{c}\begin{array}{c}\hfill r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{c}\hfill {C}_{4}\hfill \\ \hfill {C}_{2}\hfill \\ \hfill A\hfill \\ \hfill {C}_{1}\hfill \\ \hfill {C}_{3}\hfill \end{array}\right)=r\left(\begin{array}{c}\hfill A\hfill \\ \hfill {C}_{1}\hfill \\ \hfill {C}_{3}\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right),\hfill \end{array}\\ r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill Z{C}_{1}\hfill \\ \hfill {B}_{2}X\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)\le r\left(\begin{array}{cc}\hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)\end{array}$
(16)

Combining (5) with (13)-(16) yields (12).

Corollary 2.4 Let P(V1, V2, V3, V4) be given as (4) and let $R\left({B}_{1}\right)\subseteq R\left({B}_{2}\right),R\left({B}_{3}\right)\subseteq R\left({B}_{4}\right),R\left({C}_{2}^{*}\right)\subseteq R\left({C}_{1}^{*}\right),R\left({C}_{4}^{*}\right)\subseteq R\left({C}_{3}^{*}\right)$ then the matrix equation A = B1V1C1 + B2V2C2 + B3V3C 3 + B4V4C4 holds for any V1, V2, V3, and V4 if and only if τ1 = O or τ2 = O or τ3 = O.

In the rest of this section, we will find the minimal rank of the linear matrix expression P(V1,V2, V3, V4) in (4), with respect to four variant matrices ${V}_{i}\in {C}^{{p}_{i}×{q}_{i}},\phantom{\rule{2.77695pt}{0ex}}i=1,2,3,4$, when P(V1, V2, V3, V4) satisfy some restrictions.

Theorem 2.5 Let P(V1, V2, V3, V4) be given as (4) and let $R\left({B}_{1}\right)\subseteq R\left({B}_{2}\right),R\left({B}_{3}\right)\subseteq R\left({B}_{4}\right),R\left({C}_{2}^{*}\right)\subseteq R\left({C}_{1}^{*}\right),R\left({C}_{4}^{*}\right)\subseteq R\left({C}_{3}^{*}\right)$. Then

(17)

where

$\begin{array}{c}\begin{array}{c}\hfill {\beta }_{1}=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill {B}_{3}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)\right\},\hfill \end{array}\\ {\beta }_{2}=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill {C}_{4}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)\right\},\\ {\beta }_{3}=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)\right\}.\end{array}$

Proof. From the proof of Theorem 2.1, it is easy to verify that the minimal rank of P(V1, V2, V3, V4) in (4) can be expressed as

$\underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{min}}r\left(P\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)\right)=\underset{{V}_{1},{V}_{2},{V}_{3},{V}_{4}}{\text{min}}r\left(T\right)-\sum _{i=1}^{4}{p}_{i}-\sum _{i=1}^{4}{q}_{i},$
(18)

where T, S, E i , p i and q i , i = 1,2,3,4, are given as the proof of Theorem 2.1. Then applying the formula (2) in Lemma 1.1 to matrix T, we have

(19)

In this case, we derive from (19) that

(20)

Again applying the formula (2) in Lemma 1.1, we have

(21)

where S3 is given as the Equation (7) of the proof of Theorem 2.1. Since B1 = B2X, B3 = B4Y, C2 = ZC1, and C4 = WC3, (21) is reduced to

$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{min}}r\left({E}_{1}S{E}_{3}\right)\hfill \end{array}\\ \phantom{\rule{2.77695pt}{0ex}}=\sum _{i=1}^{4}{q}_{i}+{p}_{1}+{p}_{2}+r\left(A\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{B}_{2}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{B}_{4}\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)\\ \phantom{\rule{2.77695pt}{0ex}}+\text{max}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill {C}_{2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \end{array}\right),-r\left(\begin{array}{ccc}\hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\right\}\\ \phantom{\rule{2.77695pt}{0ex}}=\sum _{i=1}^{4}{q}_{i}+{p}_{1}+{p}_{2}+r\left(A\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{B}_{\text{2}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{B}_{4}\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \end{array}\right)\\ \phantom{\rule{2.77695pt}{0ex}}-r\left(\begin{array}{ccc}\hfill {C}_{2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \end{array}\right).\end{array}$
(22)

The last equality holds, since the well-known Frobenius rank inequality r(ABC) ≥ r(AB) + r(BC) − r(B), then

$\begin{array}{c}r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill Z{C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}X\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=r\left(\left(\begin{array}{cc}\hfill Z\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill I\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill X\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill I\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill I\hfill \end{array}\right)\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\ge r\left(\left(\begin{array}{cc}\hfill Z\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill I\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\right)+r\left(\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill X\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill I\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill I\hfill \end{array}\right)\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {B}_{2}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \end{array}\right).\end{array}$

With the similar method, we also have

$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{min}}r\left(\begin{array}{c}\hfill {E}_{2}\hfill \\ \hfill S\hfill \\ \hfill {E}_{4}\hfill \end{array}\right)=\sum _{i=1}^{4}{p}_{i}+{q}_{1}+{q}_{2}+r\left(\begin{array}{c}\hfill A\hfill \\ \hfill {C}_{1}\hfill \\ \hfill {C}_{3}\hfill \end{array}\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-r\left(\begin{array}{cc}\hfill {B}_{1}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill \end{array}\right)-r\left(\begin{array}{cc}\hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right),\end{array}$
(23)
$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{min}}r\left(\begin{array}{cc}\hfill O\hfill & \hfill {E}_{2}\hfill \\ \hfill {E}_{1}\hfill & \hfill S\hfill \end{array}\right)={p}_{1}+{p}_{2}+{p}_{4}+{q}_{1}+{q}_{2}+{q}_{3}+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill & \hfill {B}_{2}\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill {B}_{3}\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill \end{array}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{1}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{1}\hfill & \hfill A\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill {C}_{4}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill {C}_{2}\hfill \\ \hfill {B}_{3}\hfill & \hfill {B}_{2}\hfill & \hfill A\hfill \end{array}\right),\end{array}$
(24)
$\begin{array}{c}\begin{array}{c}\hfill \underset{{V}_{1},{V}_{2}}{\text{min}}r\left(\begin{array}{cc}\hfill S\hfill & \hfill {E}_{3}\hfill \\ \hfill {E}_{4}\hfill & \hfill O\hfill \end{array}\right)={q}_{1}+{q}_{2}+{q}_{4}+{p}_{1}+{p}_{2}+{p}_{3}+r\left(\begin{array}{ccc}\hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill {B}_{1}\hfill & \hfill A\hfill & \hfill {B}_{4}\hfill \\ \hfill O\hfill & \hfill {C}_{1}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {C}_{3}\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill {C}_{2}\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill {B}_{2}\hfill & \hfill {B}_{4}\hfill \\ \hfill {C}_{3}\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right).\end{array}$
(25)

On the other hand, by the formula (1) in Lemma 1.1, we have

(26)
(27)
$\begin{array}{c}\underset{{V}_{1},{V}_{2}}{\text{max}}r\left({E}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}S\right)={q}_{3}+{p}_{1}+{p}_{2}+{q}_{1}+{q}_{2}+\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill {C}_{4}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {C}_{2}\hfill & \hfill O\hfill \\ \hfill {}_{}\hfill \end{array}\right)\right\\end{array}$