On the determinant of certain strictly dissipative matrices (Q579363)

Let H and K be Hermitian matrices of order n with K positive definite. Then \(M=H+iK\) is said to be strictly dissipative. If H and K have eigenvalues \(\alpha_ l\geq...\geq \alpha_ n\geq 0\) and \(\beta_ l\geq...\geq \beta_ n>0\), respectively, it is shown that \[ \prod^{n}_{j=1}| \alpha_ j+i\beta_ j| \leq | \det M| \leq \prod^{n}_{j=1}| \alpha_ j+i\beta_{n-j+l}|, \] with left (right) equality if and only if M is normal with eigenvalues \(\alpha_ j+i\beta_ j\quad (\alpha_ j+i\beta_{n-j+l}).\) The authors also find bounds for the argument of the determinant of certain strictly dissipative matrices.