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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)$$ for $1\leq k \leq n$ and (singular/eigen)values sorted in the decreasing order. |
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Singular values of $X+iY$ where $X$ and $Y$ are HermitianLets have two Hermitian $n\times n$ matrices $X$ and $Y$. Are there any known properties of the singular values of $$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 $Y$.
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