Fischer type determinantal inequalities for accretive-dissipative matrices

TL;DR

This paper establishes new inequalities for the determinants of accretive-dissipative matrices, improving existing bounds by deriving sharper Fischer type inequalities based on matrix block structures.

Contribution

It introduces improved Fischer type determinant inequalities for accretive-dissipative matrices, refining previous results by Ikramov with explicit bounds depending on matrix block sizes.

Findings

Derived new bounds for |det A| in terms of submatrix determinants.

Improved upon previous inequalities by Ikramov.

Provided explicit constants depending on matrix block sizes.

Abstract

Let $A={bmatrix} A_{11} &A_{12} A_{21} & A_{22} {bmatrix}$ be an $n\times n$ accretive-dissipative matrix, $k$ and l be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$|\det A|\le a|\det A_{11}|\cdot|\det A_{22}|,$$ where $a=\{{array}{l l} 2^{3m/2}, & \text{if} m\le n/3; 2^{n/2}, & \text{if} n/3<m\le n/2. {array}.$ This improves a result of Ikramov.