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Determinantal Inequalities for Accretive-Dissipative Matrices

Authors:	Kh D Ikramov

Abstract:	A matrix $A \in M_n (C)$ is said to be accretive-dissipative if, in its Hermitian decomposition $A = B + iC,{\text{ }}B = B^* ,{\text{ C = C}}^{\text{}}$ , 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 Hermitian positive-definite matrices is proved. Let $A = \left( \begin{gathered} A_{11} {\text{ }}A_{12} \hfill \\ A_{21} {\text{ }}A_{22} \hfill \\ \end{gathered} \right)$ be an accretive-dissipative matrix, k and l be the orders of A ₁₁ and A ₂₂, respectively, and let m = min{k,l}*. Then $\|{\text{det }}A\| \leqslant 3^m \|{\text{det }}A_{11} \|\|{\text{det }}A_{22} \|$ For Buckley matrices, the stronger bound $\|{\text{det }}A\| \leqslant \left( {\frac{{1 + \sqrt {17} }}{4}} \right)^m \|{\text{det }}A_{11} \|\|{\text{det }}A_{22} \|$ is obtained. Bibliography: 5 titles.

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