Lets have two Hermitian $n\times n$ matrices $X$ and $Y$.
Are there any known properties of the singular values of $$Z = X + i Y$$ which are not generic?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)$$for 1\leq k \leq n and (singular/eigen)values sorted in the decreasing order. 1 # Singular values of X+iY where X and Y are Hermitian Lets have two Hermitian n\times n matrices X and Y. Are there any known properties of the singular values of$$Z = X + i Y which are not generic?
I am the most interested in bounding from above a few first singular values of $Z$ by the eigenvalues of $X$ and $Y$.