Lets have two Hermitian $n\times n$ matrices $X$ and $Y$.


Are there any known properties of the singular values of
$$Z = X + i Y.$$


I am the most interested in bounding from above a few first singular values of $Z$ by the eigenvalues of $X$ and $Y$. And sth that is stronger than:


$$\sum\_{i=1}^k \sigma\_i^2(Z)\leq \sum\_{i=1}^k \left( \lambda\_i^2(X)+\lambda\_i^2(Y)+\lambda\_i(i[X,Y]) \right)$$
for $1\leq k \leq n$ and (singular/eigen)values sorted in the decreasing order.



Singular values of $X+iY$ where $X$ and $Y$ are Hermitian
Some results in this direction that you might find useful are listed below.



> 
> **Theorem** (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question above. Then,
>  \begin{equation\*}
> \| (X^2+Y^2)^{1/2} \|\_p \le \|Z\|\_p \le 2^{1/2-1/p}\| (X^2+Y^2)^{1/2} \|\_p,
> \end{equation\*}
>  where $2 \le p \le \infty$, and $\|\cdot\|\_p$ denotes the Schatten-$p$ norm. The inequality above gets reversed for $1\le p \le 2$. Also, these inequalities are sharp.
> 
> 
> 


Even more directly relevant is the following theorem that discusses majorization of singular values of $X+Y$ by those of $Z$.



> 
> **Theorem** (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question. Then
>  \begin{equation\*}
> \sigma(X+Y)\quad \prec\_w\quad \sqrt{2}\sigma(Z)
> \end{equation\*}
>  If $X$ is psd, then the above weak majorization can be replaced by *weak log-majorization*, that is,
>  \begin{equation\*}
> \prod\_{j=1}^k \sigma\_j(X+Y) \le \prod\_{j=1}^k\sqrt{2}\sigma\_j(Z).
> \end{equation\*}
>  Finally, if both $X$ and $Y$ are psd, then we have even stronger inequalities:
>  \begin{equation\*}
>  \sigma\_j(X+Y) \le \sqrt{2}\sigma\_j(Z)\quad 1 \le j \le n.
> \end{equation\*}
> 
> 
> 


Bhatia and Kittaneh also discuss some applications of the above theorem to commutator inequalities.


**References**


1. R. Bhatia and F. Kittaneh. "The singular values of $A+B$ and $A+iB$." *Linear Algebra and its Applications*, 431(2009), pp. 1502-1508.
2. R. Bhatia and F. Kittaneh. "Cartesian decompositions and Schatten norms." *Linear Algebra and its Applications*, 318(2000), pp. 109--116.


