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With my basic linear algebraic background, am trying to connect the Reproducing Kernel Hilbert Space (RKHS) concepts to example functions over matrices, as I gradually learn about this new concept.

Question: Can $TrX^TKX$ be connected to a RKHS in any way? $K$ is a p.s.d kernel matrix and $X$ is a matrix of reals.

I know that $TrX^TKX$ can be represented as $Tr[(SX)^T(SX)]=||SX||^2_{HS}$, using the hilbert schmidt norm where $S$ is the p.s.d square root of K.(ex:$S=U\lambda^{1/2}$, where $K=U\lambda U^T$ is the eigen-decomposition of $K$). But am trying really hard with my narrow perspective to see the connections of this matrix function with the concept of RKHS. Do shed some light!

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