Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B

Abstract Given a complex matrix equation AXA ∗ = B , where B ∗ = ± B , we present explicit formulas for the maximal and minimal ranks of Hermitian (skew-Hermitian) solutions X to the equation as well as the maximal and minimal ranks of the real matrices X 0 and X 1 in a Hermitian (skew-Hermitian) solution X = X 0 + iX 1 . As applications, we give the maximal and minimal ranks of the real matrices C and D in a Hermitian (skew-Hermitian) g -inverse ( A + iB ) - = C + iD of a Hermitian (skew-Hermitian) matrix A + iB .

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