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I have the following trace term:

trace(Sk' Ck Sk)

where Sk is a KxM matrix and Ck is a KxK positive semidefinite matrix. I'm involving this trace term in a Matlab code and it seems to suffer when computing this term. How could I simplify it (if possible)?

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  • $\begingroup$ Which is bigger, K or M? $\endgroup$ – Omnomnomnom Mar 31 '19 at 13:21
  • $\begingroup$ K is far more smaller than M $\endgroup$ – Alberto Mar 31 '19 at 13:22
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In general, note that $\operatorname{trace}(AB) = \operatorname{trace}(BA)$, whenever both matrix multiplications are defined.

So, we can rewrite your expression as $$ \operatorname{trace}(S_k' C_k S_k) = \operatorname{trace}(S_k S_k' C_k) $$ Note that $S_k' C_k S_k$ is $M \times M$, whereas $S_k S_k' C_k$ is $K \times K$.

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  • $\begingroup$ That's true, I did not considered to shift the matrices... but it seems to make a lighter problem in Matlab . Thank you $\endgroup$ – Alberto Mar 31 '19 at 14:18

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