Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA*=B with applications
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Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.
KeywordsMaximal rank Minimal rank Submatrices Hermitian solution Skew-Hermitian solution matrix equation
Mathematics Subject Classification (2000)15A03 15A09 15A24
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