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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.

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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 $$\text{Tr}( X^T A X) =\text{Tr}( X^T C^T C X) =\text{Tr}[ (CX)^T (C X)] = \lVert CX \rVert_\text{HS}^2. $$

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It doesn't have to be Cholesky: any other way to write $A = C^T C$ will do just as well. For example, $C$ could be the positive semidefinite square root of $A$. –  Robert Israel Aug 16 '12 at 23:37
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