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Is there a spectral or eigen-value solution to finding $X$ such that $Tr(CX^TMX)$ is minimum for a symmetric matrix $C$ and a p.s.d matrix $M$. Also there is a linear constraint on the minimization that $TrX^TL=\lambda$ where $L$ is a given real matrix and $\lambda$ is a given real scalar.

If not what would be some solution to find $X$ that gives the minimum?

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