Determinantal inequalities for accretive-dissipative matrices

Kh. D. Ikramov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XV, Tome 284 (2002), pp. 36-47

Résumé

A matrix $A\in M_n(\mathbf C)$ is said to be accretive-dissipative if in its Hermitian decomposition $$ A=B+iC, \quad B=B^*, \quad C=C^*, $$ both matrices $B$ and $C$ are positive definite. Further, if $B=I_n$, then $A$ is called a Buckley matrix. The following extension of the classical Fischer inequality for Hermtian positive-definite matrices is proved. Let \begin{math} A=\begin{pmatrix} A_{11}{12} A_{21}{22} \end{pmatrix} \end{math} be an accritive-dissipative matrix, $k$ and $l$ be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$ |{\det A}|\le3^m|{\det A_{11}}|\,|{\det A_{22}}|. $$ For Buckley matrices, the stronger bound $$ |{\det}|\le\biggl(\frac{1+\sqrt{17}}4\biggr)^m|{\det A_{11}}|\,|{\det A_{22}}|. $$ is obtained.