Criterion for a real square matrix to have a Volterra multiplier (Q2721551)

A real square matrix \(A\) is strictly accretive if \(A+A^T>0\). If \(D\) is a diagonal positive matrix, then \(A\) is said to have \(D\) as a Volterra multiplier when \(DA+(DA)^T >0\). In this note, the authors state without proof the simple facts that \(A=\left[\begin{smallmatrix} A_1 & A_2\\ A_3 & A_4 \end{smallmatrix} \right]\) is strictly accretive (resp., has a Volterra multiplier) if and only if \(A_1\) and \(A_4-1/2 (A_3+A_2^T)\) \((A_1+A^T_1)^{-1} (A_2+A_3^T)\) are strictly accretive (resp., \(A_1\) and, for some diagonal positive matrices \(D_1\) and \(D_2\), \(A_4-1/2 (A_3+D_2^{-1} A_2^TD_1)\) \((D_1A_1 +(D_1A_1)^T)^{-1} (D_1A_2+ (D_2A_3)^T)\) have Volterra multipliers).