New properties of a special matrix related to positive-definite matrices

Abstract

Let $H$ be a $2n\times 2n$ real symmetric positive-definite matrix. Suppose that $H\circ H=(H_{ij})_{2n\times 2n}$ is a partitioned matrix, in which $\circ$ represents the Hadamard product and the block $H_{ij}$ has order $n\times n$, $1\leq i,j \leq 2$. Several new properties on the matrix $\widetilde{H}$ are derived including inequalities that involve the symplectic eigenvalues and the usual eigenvalues, where $2\widetilde{H}=H_{11}+H_{22}+H_{12}+H_{21}$.