Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B

This work is devoted to establish the extremal inertias ofthe two submatrices $X_{1}$ and $X_{4}$ in a Hermitian least rank solution $X$of the matrix equation $AXA^{*}=B$. From these formulas, necessary andsufficient conditions for these submatrices to be positive (nonpositive,negative, nonnegative) definite are achieved.


SIHEM GUERARRA
solving ranks and inertias minimization problems associated with matrix equations and their solutions. For example see [4], [5], such that in the first one, the authors studied the extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA * = B, in the second work, the authors studied the ranks of Hermitian and skew Hermitian solutions to (1). Also in [10] Y. Tian gave the necessary and sufficient conditions for the least squares solutions and least rank solutions of (1) to coincide, in [3] the present author and S. Guedjiba gave the necessary and sufficient conditions for the paire of matrix equations A 1 XA * 1 = B 1 and A 2 XA * 2 = B 2 to have a common Hermitian positive definite, negative definite, nonpositive definite, nonnegative definite least rank solution.
The concept of least-rank solutions of matrix equations was proposed in [7] and [11] by Y. Tian in studying the minimal rank of the linear matrix function A−BXC.
In [10] the Hermitian least rank solution of (1) is the matrix X which minimizes the rank of the difference B − AXA * or equivalently The Hermitian least-rank solution of (1) is the solution of the consistent equation Equation (3) is called the normal equation associated with (2). Hence the general expression of the Hermitian least rank solution of (1) can be written by where M = B A A * 0 , T = 0 I n , T 1 = T F M , and U ∈ C (m+n)×n is arbitrary. We need the following Lemmas.
[9] Let S be a set consisting of matrices over C m×n , and let be a set consisting of Hermitian matrices over C m×m . Then, a) has a matrix X > 0, (X < 0) if and only if max Following the work of Y. Tian in [9], in which the author derived necessary and sufficient conditions for the Hermitian least squares solution of the matrix equation (1) to have a submatrices positive or negative definites, in this work we derive these results on the other solution of this equation which is the Hermitian least rank solution, also to have a submatrices positive (negative, nonpositive, nonnegative) definites.
2. Positive and negative definite submatrices in an Hermitian solution of AXA * = B. For convenience of representation, the following notation for the collection of Hermitian least rank solutions of equation (1) is adopted The general least rank solutions of (1) are given by where We note that, if B * = B, then M * = M is Hermitian also. One of the fundamental concepts in matrix theory is the partition of matrix, many properties of a matrix can be drived from the submatrices in its partition. In order to show more properties of Hermitian least rank solution of Eq (1), the Hermitian least rank solution X ∈ S in Eq (1) is partitioned into 2 × 2 block form So, Eq (1) can be written as where , with n 1 + n 2 = n. It is easy to see that X 1 , X 2 , X 3 and X 4 can be written as Substituting its Hermitian least rank solution into above formulas yields the general expressions of X 1 , X 2 , X 3 and X 4 where U = U 1 U 2 . in this work we wont to drive the extremal inertias of the Hermitian two submatrices X 1 and X 4 , so we adopt the following notations for the collections of submatrices X 1 , X 4 in (11) where A ∈ C m H , B ∈ C m×n are given and X ∈ C n×m is a variable matrix, and let N =
Next, we achieve at the necessary and sufficient conditions of the submatrices X 1 and X 4 to be positive (negative, nonpositive, nonnegative) definite, from Theorem 2.2 and Lemma 1.4.