Determinantal inequalities for block triangular matrices

TL;DR

This paper establishes new determinantal inequalities for block triangular matrices, showing how the determinant of a block matrix relates to its diagonal blocks, with conditions for equality.

Contribution

It introduces novel determinantal inequalities for block triangular matrices and characterizes conditions for equality, extending existing matrix inequality theory.

Findings

Proves $ ext{det}(I_n + T^*T) igg floor ext{det}(I_r + X^*X) imes ext{det}(I_{n-r} + Z^*Z)$.

Equality holds if and only if the off-diagonal block $Y$ is zero.

Provides a new inequality framework for block triangular matrices.

Abstract

Let $T=\begin{bmatrix} X &Y\\ 0 & Z\end{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $\det(I_n+T^*T)\ge \det(I_r+X^*X)\cdot \det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.