Upper and Lower Bounds for Ranks of the Matrix Expression X − XAX

and Applied Analysis 3


Introduction
Throughout this paper  × denotes the set of all  ×  matrices over the complex field .  denotes the identity matrix of order  and  × is the  ×  matrix of all zero entries (if no confusion occurs, we will drop the subscript).For a matrix  ∈  × ,  * and () denote the conjugate transpose and the rank of the matrix , respectively.(, ) denotes a row block matrix consisting of  ∈  × and  ∈  × .
In matrix theory and applications, there exists a nonlinear matrix expression that involves variable entries: where  ∈  × is a given complex matrix and  ∈  × is a variable matrix.These nonlinear matrix expressions vary with respect to the choice of .One of the fundamental problems for (2) is to determine the maximal and minimal possible ranks of the matrix expression () when  is running over  × .Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [5], then the maximal and minimal ranks of () can be attained for some .
The investigation of extremal ranks of matrix expressions has many direct motivations in matrix analysis.For example, a matrix expression  −  of order  is nonsingular if and only if the maximal rank of  −  with respect to  is ; two consistent matrix equations  1 =  1  1 and  2 =  2  2 have a common solution if and only if the minimal rank of the difference  1 −  2 of their solutions is zero; a nonlinear matrix equation  =  is consistent if and only if the minimal rank of  −  with respect to  is zero.From the definition of the {2}-inverse of a matrix, we know that the solution  of the nonlinear matrix equation  =  is a {2}-inverse of matrix ; that is, using the minimal rank of  − , we can find out the general expression of the {2}-inverses of a matrix , which is a matrix such that the nonlinear matrix expression  −  attains its minimal rank.In general, for any two matrix expressions ( These examples imply that the extremal ranks of matrix expressions have close links with many topics in matrix analysis and applications.Various statements on maximal and minimal ranks of matrix expressions are quite easy to understand for the people who know linear algebra.But the question now is how to give simple or closed forms for the extremal ranks of a matrix expression with respect to its variant matrices.The study on maximal and minimal ranks of matrix expression started in late 1980s.If want to know more about this question the reader can see [6][7][8][9][10][11][12][13][14][15][16][17][18].
The work in this paper includes two parts.First, in Section 2, we will consider how to choose a matrix  ∈  × , such that  −  has the maximal possible rank.Second, in Section 3, we will determine the minimal rank of − and present a general expression of the {2}-inverses of matrix  ∈  × .
In order to find the extremal ranks of the nonlinear matrix expression  − , we need the following lemmas, which will be used in this paper.
, and  2 ∈   2 × are given matrices.Then for any variable matrices min where Lemma 2 (see [20]).Let − be a linear matrix expression over the complex field , where  ∈ the general expression of  satisfying (7) is where

The Maximal Rank of 𝑋−𝑋𝐴𝑋 with respect to 𝑋
Let  ∈  × be a given matrix; in this section, we will present the maximal rank of the nonlinear matrix expression  − , with respect to the variable matrix  ∈  × .The relative results are included in the following three lemmas.
Proof.First describe a special congruence transformation for a block matrix, which reduces the calculation of the maximal rank of  − : (37)

The Minimal Rank of 𝑋−𝑋𝐴𝑋 with respect to 𝑋
In this section, we will present the minimal rank of the nonlinear matrix expression  − .Moreover, we will consider how to choose a matrix , such that  −  has the minimal possible rank.

) Lemma 6 .
Let  ∈  × , and  =   denotes the identity matrix of order .Then max
∈  × is a variant matrix.Then the maximal rank of  −  with respect to  is ×,  ∈  × , and  ∈  × are given;