Define $f(X) = \operatorname{tr}(MXX^T)$. If $M$ is a positive semi-definite matrix, can we prove that $f$ is convex?
Can we prove $\operatorname{tr}(M X X^T)$ is convex?
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linear-algebra
convex-analysis
1 Answers
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$ s\cdot f(X)+(1-s)\cdot f(Y)=f(sX+(1-s)Y)\color{red}{+s(1-s)\cdot f(X-Y)} $
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0$M(sX+(1-s)Y)(sX+(1-s)Y)^T=\ldots$ and the operator trace is linear hence $f(sX+(1-s)Y)=\ldots$ – 2012-03-23