Some singular value inequalities via convexity

TL;DR

This paper presents a simple convexity-based proof of a fundamental singular value inequality involving Hadamard products, and extends the approach to derive additional related inequalities.

Contribution

It introduces a straightforward convexity argument to prove a key singular value inequality and applies this technique to establish new related inequalities.

Findings

Provided a simple proof of a fundamental singular value inequality.

Extended the convexity approach to derive additional inequalities.

Demonstrated the effectiveness of convexity arguments in matrix analysis.

Abstract

If $c_1(Z) \geq ... \geq c_n(Z)$ denote the Euclidean lengths of the column vectors of any $n \times n$ matrix $Z,$ then a fundamental inequality related to Hadamard products states that $$ \sum_{i=1}^k \sigma_i(X^*Y \circ B) \leq \sum_{i=1}^k c_i(X) c_i(Y) \sigma_i(B) \qquad 1 \leq k \leq n,$$ where $\sigma_i(\cdot)$ is the $i$th singular value. In this paper, we shall offer a simple proof of this result via convexity arguments. In addition, this technique is applied to obtain some further singular value inequalities as well.