Singular Value Inequalities for Real and Imaginary Parts of Matrices

Abstract

Let $A = {\rm Re} \ A + i \ {\rm Im} \ A$ be the Cartesian decomposition of square matrix $A$ of order $n$ with ${\rm Re} \ A = \frac{A+A^*}{2}
$ ${\rm Im} \ A = \frac{A-A^*}{2i}$. Fan-Hoffman's result asserts that
$$
\lambda_{j}({\rm Re} A) \leq s_{j}(A), \ j =1,\ldots,n,
$$
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:
\begin{equation*}
s_{j}({\rm Re} \ A) \leq \frac{1}{4}s_{j}\left([(|A|+|A^*|)-(A+A^*)]\oplus [(|A|+|A^*|)+(A+A^*)]\right),
\end{equation*}
and
\begin{equation*}
s_{j}({\rm Im} \ A) \leq \frac{1}{4}s_{j}\left([(|A|+|A^*|)-i(A^*-A)]\oplus [(|A|+|A^*|)+i(A^*-A)]\right), \ j =1,\ldots,n.
\end{equation*}
 In particular, we have
\begin{equation*}
s_{j}({\rm Re} \ A) \leq \frac{1}{2}s_{j}\left((|A|+|A^*|)\oplus (|A|+|A^*|)\right),
\end{equation*}
and
\begin{equation*}
s_{j}({\rm Im} \ A) \leq \frac{1}{2}s_{j}\left((|A|+|A^*|)\oplus (|A|+|A^*|)\right),  \ j=1,\ldots,n.
\end{equation*}
Moreover, we also show that these inequalities are sharp.