Singular Value Inequalities for Real and Imaginary Parts of Matrices

Dongjun Chen, Yun Zhang

Let $A=\operatorname{Re}A+i$ $\operatorname{Im}A$ be the Cartesian decomposition of square matrix $A$ of order $n$ with $\operatorname{Re}A=\frac{A+A^*}2$ and $\operatorname{Im}A=\frac{A-A^*}{2i}$. Fan--Hoffman's result asserts that \[ ambda_j(peratorname{Re}A)eq s_j(A),\quad j=1,\dots,n, \] where $\lambda j(M)$ and $s_j(M)$ stand for the $j$th largest eigenvalue of $M$ and the $j$th largest singular value of $M$, respectively. We investigate singular value inequalities for real and imaginary parts of matrices and prove the following inequalities: \[ s_j(peratorname{Re}A)eqfrac14s_j([(|A|+|A^*|)-(A+A^*)]times[(|A|+|A^*|)+(A+A^*)]), \] and \[ s_j(peratorname{Im}A)eqfrac14s_j([(|A|+|A^*|)-i(A^*-A)]times[(|A|+|A^*|)+i(A^*+A)]),\quad j=1,\dots,n. \] In particular, we have \[ s_j(peratorname{Re}A)eqfrac12s_j((|A|+|A^*|)times(|A|+|A^*|)), \] and \[ s_j(peratorname{Im}A)eqfrac12s_j((|A|+|A^*|)times(|A|+|A^*|)),\quad j=1,\dots,n. \] Moreover, we also show that these inequalities are sharp.