Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals and $A$ is a positive semi-definite matrix. Am reading through a fundamental book in Functional analysis and hilbert spaces, and have a background in Linear Algebra and Advanced Calculus. Thank you.
2026-04-02 04:47:08.1775105228
Representing with Hilbert Schmidt Norm
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In order to be able to express $\text{Tr}(X^T A X)$ as a Hilber-Schmidt Norm you need to find the Cholesky decomposion of $A$ in terms of $C$, $A= C^T C$. Given that $$\text{Tr}( X^T A X) =\text{Tr}( X^T C^T C X) =\text{Tr}[ (CX)^T (C X)] = \lVert CX \rVert_\text{HS}^2. $$