# trace function, eigen decomposition and optimization!

The equation \begin{align} \min_{X}~trace(CX^{T}MX) \end{align} where $C$ is symmetric and M is symmetric , p.s.d can be minimized by defining $M=F^{T}F$ ($M$ being a psd matrix, you will be able to find such a $F$)

and defining $Y=FX$,

we get the above problem as minimizing \begin{align} \min_{Y}~trace(CY^{T}Y) \end{align}

Now if $Y^*$ is the optimal solution would $X^*=YF^{-1}$ be the optimal solution for $X$ for minimizing $trace(CX^{T}MX)$ as we defined $Y$ as $Y=FX$?

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If $Y=FX$ then multiply on the left by $F^{-1}$ to get $F^{-1}Y=X$, matrix multiplication is not commutative.