The Re-nonnegative definite solutions to the matrix inverse problem \(AX=B\)
Let \(A\) be an \(n\times n\) complex matrix. Then \(A\) is called ``Re-nonnegative definite'' (Re-nnd) if the real part of \(x* Ax\) is nonnegative for all complex \(n\)-vectors \(x\) (equivalently, if \(H(A):= {1\over 2} (A+ A*)\) is nonnegative definite). Analogous conditions define ``Re-positive definite matrices'' (Re-pd). The primary result of the paper is a necessary and sufficient condition for there to be at least one Re-nnd matrix \(A\) satisfying \(AX =B\) for given \(n\times m\) matrices \(X\) and \(B\). Theorem 5. An \(n\times n\) Re-nnd matrix \(A\) exists such that \(AX= B\) if and only if (i) there exists an \(n\times n\) matrix \(A_0\) such that \(A_0 X=B\) and (ii) \(X * B\) is Re-nnd. This result is analogous to a theorem for Re-pd matrices proved earlier by \textit{L. Wu} [Linear Algebra Appl. 174, 145-151 (1992; Zbl 0754.15003)].
scientific article; zbMATH DE number 871263