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The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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How to design a matrix function to meet the following conditions

There is no such function Assumption on the Function space I assume that $f$ has to be a polynomial in the variables $X$ and $Q$, i.e., $f$ is a sum of terms of the form $$ a_k X^{k_1} Q^{k_2} X^{k_3} …
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