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 C^{m × 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 ∈ C^{m× 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 ∈ C^{n× m}of A ∈ C^{m × n}is a matrix which satisfies some of the following four Penrose equations [1]:

$$\left(1\right)AXA=A,\left(2\right)XAX=X,\left(3\right){\left(AX\right)}^{*}=AX,\left(4\right){\left(XA\right)}^{*}=XA.$$

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 − A^†A 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 [4–6]; 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 [7–11].

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, 10–16]. 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 [17–20]. 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, 21–28].

In this article, given matrices $A\in C^{m\times n},B_i\in C^{m\times p_i},C_i\in C^{q_i\times n},i=1,2,3,4$, 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., [4–6, 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 $A_{ij}\in C^{m_i\times n_j}\left(1\le i,j\le 3\right)$ are given, $X\in C^{m_1\times n_3}$ and $Y\in C^{m_3\times n_1}$ are two variant matrices. Then

$$\begin{array}{c}\underset{X,Y}{\text{max}}r\left(M\right)=\text{min}\left\{m_3+n_3+r\left(\begin{array}{cc}\hfill A_{11}\hfill & \hfill A_{12}\hfill \\ \hfill A_{21}\hfill & \hfill A_{22}\hfill \end{array}\right),m_1+n_1+r\left(\begin{array}{cc}\hfill A_{22}\hfill & \hfill A_{23}\hfill \\ \hfill A_{32}\hfill & \hfill A_{33}\hfill \end{array}\right),\right.\\ \left.m_1+m_3+r\left(A_{21}{A}_{22}{A}_{23}\right),n_1+n_3+r\left(\begin{array}{c}\hfill A_{12}\hfill \\ \hfill A_{22}\hfill \\ \hfill A_{32}\hfill \end{array}\right)\right\},\end{array}$$

(1)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{X,Y}{\text{min r}}\left(M\right)=r\left(A_{21}{A}_{22}{A}_{23}\right)+r\left(\begin{array}{c}\hfill A_{12}\hfill \\ \hfill A_{22}\hfill \\ \hfill A_{32}\hfill \end{array}\right)+\text{max}\left\{r\left(\begin{array}{cc}\hfill A_{11}\hfill & \hfill A_{12}\hfill \\ \hfill A_{21}\hfill & \hfill A_{22}\hfill \end{array}\right)-r\left(\begin{array}{c}\hfill A_{12}\hfill \\ \hfill A_{22}\hfill \end{array}\right)-r\left(A_{21}{A}_{22}\right),\right.\hfill \end{array}\\ \left.r\left(\begin{array}{cc}\hfill A_{22}\hfill & \hfill A_{23}\hfill \\ \hfill A_{32}\hfill & \hfill A_{33}\hfill \end{array}\right)-r\left(\begin{array}{c}\hfill A_{22}\hfill \\ \hfill A_{32}\hfill \end{array}\right)-r\left(A_{22}{A}_{23}\right)\right\}.\end{array}$$

(2)

Lemma 1.2 [2]. Let A ∈ C^{m × 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 ∈ C^{n × m}and Z ∈ C^{n× m}are arbitrary.

Lemma 1.3 [9]. Let A ∈ C^{m × n}, B ∈ C^{m × k}, and C ∈ C^{l × 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 - A^†A.

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 $A\in C^{m\times n},B_i\in C^{m\times p_i},C_i\in C^{q_i\times n},i=1,2,3,4$, are given matrices, with respect to four variant matrices $V_i\in C^{p_i\times q_i},i=1,2,3,4$. 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 $A\in C^{m\times n},B_i\in C^{m\times p_i},C_i\in C^{q_i\times n},V_i\in C^{p_i\times q_i},i=1,2,3,4$ 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

$$\begin{array}{c}{E}_1={\left(OOOOOO{I}_{q_3}\right)}^{*},E_{\text{2}}=\left(OOOOOO{I}_{p_4}\right),\\ E_3={\left(I_{q_4}OOOOOO\right)}^{*},E_4=\left(I_{p_3}OOOOOO\right).\end{array}$$

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

$$\begin{array}{c}\underset{V_3,V_4}{\text{max}}r\left(T\right)=\text{min}\left\{p_3+q_4+r\left(\begin{array}{cc}\hfill O\hfill & \hfill E_2\hfill \\ \hfill E_1\hfill & \hfill S\hfill \end{array}\right)p_4+q_3+r\left(\begin{array}{cc}\hfill S\hfill & \hfill E_3\hfill \\ \hfill E_4\hfill & \hfill O\hfill \end{array}\right)\right.,\\ \left.p_4+p_3+r\left(E_{1}S{E}_2\right),q_3+q_4+r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)\right\}\\ =\text{min}\left\{p_3+q_4+p_4+q_3+r\left(S_1\right),p_4+q_3+q_4+p_3+\right.r\left(S_2\right),\\ \left.p_4+p_3+q_4+q_3+r\left(S_3\right),q_3+q_4+p_4+p_3+r\left(S_4\right)\right\},\end{array}$$

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

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2,V_3,V_4}{\text{max}}r\left(T\right)=\text{min}\left\{p_3+q_4+p_4+q_3+\underset{V_1,V_2}{\text{max}}r\left(S_1\right),p_4+q_3+q_4+p_3+\underset{V_1,V_2}{\text{max}}r\left(S_2\right)\right.,\hfill \end{array}\\ \left.p_4+p_3+q_4+q_3+\underset{V_1,V_2}{\text{max}}r\left(S_3\right),q_3+q_4+p_4+p_3+\underset{V_1,V_2}{\text{max}}r\left(S4\right)\right\}\end{array}$$

(7)

and

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{max}}r\left(S_1\right)=\text{min}\left\{p_1+q_2+q_1+p_2+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill \end{array}\right),p_1+q_2+q_1+p_2+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right),\right.\hfill \end{array}\\ \left.p_1+q_2+q_1+p_2+r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_2\hfill \end{array}\right),p_1+q_2+q_1+p_2+r\left(\begin{array}{cc}\hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_1\hfill \end{array}\right)\right\},\end{array}$$

(8)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{max}}r\left(S_2\right)=\text{min}\left\{p_1+q_2+q_1+p_2+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill \end{array}\right),p_1+q_2+q_1+p_2+r\left(\begin{array}{ccc}\hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill C_3\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),\right.\hfill \end{array}\\ \left.p_1+q_2+q_1+p_2+r\left(\begin{array}{cccc}\hfill B_1\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),p_1+q_2+q_1+p_2+r\left(\begin{array}{cc}\hfill C_2\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right)\right\},\end{array}$$

(9)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{max}}r\left(S_3\right)=\text{min}\left\{p_1+q_2+q_1+p_2+r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \end{array}\right),\right.\hfill \end{array}\\ p_1+q_2+q_1+p_2+r\left(\begin{array}{cccc}\hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),\\ p_1+q_2+q_1+p_2+r\left(B_3{B}_{1}A{B}_2{B}_4\right),\\ p_1+q_2+q_1+p_2+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right),\end{array}$$

(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), $R(B_1)\subseteq R(B_2),R(B_3)\subseteq R(B_4),R(C_2^{\ast })\subseteq R(C_1^{\ast }),R(C_4^{\ast })\subseteq R(C_3^{\ast })$ 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

$$\begin{array}{c}{\tau }_1=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill O\hfill & \hfill C_4\hfill \\ \hfill B_3\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_1\hfill \end{array}\right)\right\},\\ {\tau }_2=\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \end{array}\right),r\left(A{B}_2{B}_4\right),r\left(\begin{array}{cc}\hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \end{array}\right)\right\},\\ {\tau }_3=\text{min}\left\{r\left(\begin{array}{cc}\hfill O\hfill & \hfill C_2\hfill \\ \hfill B_1\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_3\hfill \end{array}\right),r\left(\begin{array}{cc}\hfill A\hfill & \hfill B_2\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{c}\hfill A\hfill \\ \hfill C_1\hfill \\ \hfill C_3\hfill \end{array}\right)\right\}.\end{array}$$

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 $V_i\in C^{p_i\times q_i},i=1,2,3,4$, 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

$$\begin{array}{c}\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)\\ =r\left(A{B}_{\text{2}}{B}_{\text{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}{cc}\hfill A\hfill & \hfill B_2\hfill \\ \hfill C_3\hfill & \hfill O\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)\\ +r\left(\begin{array}{c}\hfill A\hfill \\ \hfill C_1\hfill \\ \hfill C_3\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 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}{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)\\ +\text{max}\left\{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)+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)\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)-{\beta }_1-{\beta }_2,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)\\ \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 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)-2{\beta }_3\right\},\end{array}$$

(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

$$\begin{array}{c}\underset{V_1,V_2,V_3,V_4}{\text{min}}r\left(T\right)=\underset{V_3,V_4}{\text{min}}r\left(\begin{array}{ccc}\hfill O\hfill & \hfill E_2\hfill & \hfill -V_4\hfill \\ \hfill E_1\hfill & \hfill S\hfill & \hfill E_3\hfill \\ \hfill -V_3\hfill & \hfill E_4\hfill & \hfill O\hfill \end{array}\right)=r\left(E_{1}S{E}_3\right)+r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)\\ +\text{max}\left\{r\left(\begin{array}{cc}\hfill O\hfill & \hfill E_2\hfill \\ \hfill E_1\hfill & \hfill S\hfill \end{array}\right)-r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \end{array}\right)-r\left(E_{1}S\right),\right.\\ \left.r\left(\begin{array}{cc}\hfill S\hfill & \hfill E_3\hfill \\ \hfill E_4\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{c}\hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)-r\left(S{E}_3\right)\right\}.\end{array}$$

(19)

In this case, we derive from (19) that

$$\begin{array}{c}\underset{V_1,V_2,V_3,V_4}{\text{min}}r\left(T\right)=\underset{V_1,V_2}{\text{min}}r\left(E_{1}S{E}_{\text{3}}\right)+\underset{V_1,V_2}{\text{min}}r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)\\ +\text{max}\left\{\underset{V_1,V_2}{\text{min}}r\left(\begin{array}{cc}\hfill O\hfill & \hfill E_2\hfill \\ \hfill E_1\hfill & \hfill S\hfill \end{array}\right)-\underset{V_1,V_2}{\text{min}}r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \end{array}\right)-\underset{V_1,V_2}{\text{min}}r\left(E_{1}S\right),\right.\\ \left.\underset{V_1,V_2}{\text{min}}r\left(\begin{array}{cc}\hfill S\hfill & \hfill E_3\hfill \\ \hfill E_4\hfill & \hfill O\hfill \end{array}\right)-\underset{V_1,V_2}{\text{min}}r\left(\begin{array}{c}\hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)-\underset{V_1,V_2}{\text{min}}r\left(S{E}_3\right)\right\}.\end{array}$$

(20)

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

$$\begin{array}{c}\underset{V_1,V_2}{\text{min}}r\left(E_{1}S{E}_3\right)=q_4+q_3+\underset{V_1,V_2}{\text{min}}r\left(S_3\right)\\ =\sum _{i=1}^4{q}_i+p_1+p_2+r\left(B_3{B}_{\text{1}}A{B}_2{B}_4\right)+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right)\\ +\text{max}\left\{r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \end{array}\right)-r\left(\begin{array}{cccc}\hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccccc}\hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \end{array}\right),\right.\\ \left.r\left(\begin{array}{cccc}\hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{cccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccccc}\hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\right\},\end{array}$$

(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

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{min}}r\left(E_{1}S{E}_3\right)\hfill \end{array}\\ =\sum _{i=1}^4{q}_i+p_1+p_2+r\left(A{B}_2{B}_4\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \end{array}\right)\\ +\text{max}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill C_2\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \end{array}\right),-r\left(\begin{array}{ccc}\hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)\right\}\\ =\sum _{i=1}^4{q}_i+p_1+p_2+r\left(A{B}_{\text{2}}{B}_4\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \end{array}\right)+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \end{array}\right)\\ -r\left(\begin{array}{ccc}\hfill C_2\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \end{array}\right).\end{array}$$

(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)\\ =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)\\ \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)\\ -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)\\ =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

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{min}}r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)=\sum _{i=1}^4{p}_i+q_1+q_2+r\left(\begin{array}{c}\hfill A\hfill \\ \hfill C_1\hfill \\ \hfill C_3\hfill \end{array}\right)+r\left(\begin{array}{cc}\hfill A\hfill & \hfill B_2\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill O\hfill & \hfill C_2\hfill \\ \hfill B_1\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_3\hfill \end{array}\right)\\ -r\left(\begin{array}{cc}\hfill B_1\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_1\hfill \\ \hfill O\hfill & \hfill C_3\hfill \end{array}\right)-r\left(\begin{array}{cc}\hfill C_2\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_2\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right),\end{array}$$

(23)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{min}}r\left(\begin{array}{cc}\hfill O\hfill & \hfill E_2\hfill \\ \hfill E_1\hfill & \hfill S\hfill \end{array}\right)=p_1+p_2+p_4+q_1+q_2+q_3+r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill O\hfill & \hfill C_4\hfill \\ \hfill B_3\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_1\hfill \end{array}\right)\\ +r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_1\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_4\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill B_2\hfill & \hfill A\hfill \end{array}\right),\end{array}$$

(24)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{min}}r\left(\begin{array}{cc}\hfill S\hfill & \hfill E_3\hfill \\ \hfill E_4\hfill & \hfill O\hfill \end{array}\right)=q_1+q_2+q_4+p_1+p_2+p_3+r\left(\begin{array}{ccc}\hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill C_3\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right)+r\hfill \end{array}\left(\begin{array}{cc}\hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right)\\ +r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_1\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill \end{array}\right)-r\left(\begin{array}{ccc}\hfill C_2\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill C_3\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right).\end{array}$$

(25)

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

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{min}}r\left(\begin{array}{c}\hfill E_2\hfill \\ \hfill S\hfill \end{array}\right)=p_4+p_1+p_2+q_1+q_2+\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill O\hfill & \hfill C_2\hfill \\ \hfill B_3\hfill & \hfill B_1\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C_3\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill B_3\hfill & \hfill A\hfill & \hfill B_2\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill \end{array}\right),\right.\hfill \end{array}\\ \left.r\left(\begin{array}{cc}\hfill B_3\hfill & \hfill A\hfill \\ \hfill O\hfill & \hfill C_1\hfill \\ \hfill O\hfill & \hfill C_3\hfill \end{array}\right)\right\}.\end{array}$$

(26)

$$\begin{array}{c}\begin{array}{c}\hfill \underset{V_1,V_2}{\text{max}}r\left(\begin{array}{c}\hfill S\hfill \\ \hfill E_4\hfill \end{array}\right)=p_3+p_1+p_2+q_1+q_2+\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill B_1\hfill & \hfill A\hfill & \hfill B_4\hfill \\ \hfill O\hfill & \hfill C_3\hfill & \hfill O\hfill \end{array}\right),r\left(\begin{array}{ccc}\hfill C_4\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill A\hfill & \hfill B_2\hfill & \hfill B_4\hfill \\ \hfill C_3\hfill & \hfill O\hfill & \hfill O\hfill \end{array}\right),\right.\hfill \end{array}\\ r\left.\left(\begin{array}{cc}\hfill A\hfill & \hfill B_4\hfill \\ \hfill C_1\hfill & \hfill O\hfill \\ \hfill C_3\hfill & \hfill O\hfill \end{array}\right)\right\},\end{array}$$

(27)

$$\begin{array}{c}\underset{V_1,V_2}{\text{max}}r\left(E_{1}S\right)=q_3+p_1+p_2+q_1+q_2+\text{min}\left\{r\left(\begin{array}{ccc}\hfill O\hfill & \hfill C_4\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill C_2\hfill & \hfill O\hfill \\ \hfill {}_{}\hfill \end{array}\right)\right.\end{array}$$