# Can we prove $\operatorname{tr}(M X X^T)$ is convex?

Define $f(X) = \operatorname{tr}(MXX^T)$. If $M$ is a positive semi-definite matrix, can we prove that $f$ is convex?

$$s\cdot f(X)+(1-s)\cdot f(Y)=f(sX+(1-s)Y)\color{red}{+s(1-s)\cdot f(X-Y)}$$
• $M(sX+(1-s)Y)(sX+(1-s)Y)^T=\ldots$ and the operator trace is linear hence $f(sX+(1-s)Y)=\ldots$ – Did Mar 23 '12 at 18:02