The maximal and minimal ranks of matrix expression with applications

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.

where W ∈ C ^{n × m} and Z ∈ C ^{n × m}are arbitrary.

where E _A = I _m - AA^† and F _A = I _n - A^†A.

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.

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.

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.