On Drury's solution of Bhatia \& Kittaneh's question

TL;DR

This paper revisits and simplifies Drury's proof of a positive semidefinite matrix inequality originally posed by Bhatia and Kittaneh, confirming a conjecture relating singular values and eigenvalues.

Contribution

The paper provides a simplified proof of Drury's solution to a matrix inequality question posed by Bhatia and Kittaneh, enhancing understanding of the result.

Findings

Confirmed the inequality for positive semidefinite matrices.

Simplified the proof of a key auxiliary result.

Validated Drury's affirmative solution.

Abstract

Let $A, B$ be $n\times n$ positive semidefinite matrices. Bhatia and Kittaneh asked whether it is true $$ \sqrt{\sigma_j(AB)}\le \frac{1}{2} \lambda_j(A+B), \qquad j=1, \ldots, n$$ where $\sigma_j(\cdot)$, $\lambda_j(\cdot)$, are the $j$-th largest singular value, eigenvalue, respectively. The question was recently solved by Drury in the affirmative. This article revisits Drury's solution. In particular, we simplify the proof for a key auxiliary result in his solution.