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For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite. In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$, $X\ge W …

Let $H$ be an $n\times n$ Hilbert matrix, $$h_{ij}=(i+j-1)^{-1}.$$ The matrix $p$-norm corresponding to the p-norm for vectors is: $\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left \ …

Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant …

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I guesse …

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ denot …

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here). Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very l …

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, …

Let $A, B$ be Hermitian matrices. Does the following hold? $$\det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)$$ As usual, $|X|=(X^*X)^{1/2}$. Clearly, quantities on both sides are no less t …

Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is $$\lambda_{j}(X^{*}X+Y^{*}Y)\geq 2\sigma_j(XY^*)$$ for $j \in \{1, \ldots, n\}$, where $\lambda_j( …