# Is the composition of trace inverse and convex matrix product convex?

Is the trace of the inverse of the matrix product $B^TB$, i.e. $\mathrm{trace}((B^TB)^{-1})$, convex where $B\in M_{n,m}$.

I know that $S\longrightarrow \mathrm{trace}(S^{-1})$ is a convex function and you can find the proof here Is the trace of inverse matrix convex?. But now I'm asking about the composition of two convex functions which are $f: S\longrightarrow \mathrm{trace}(S^{-1})$ and $g: B\longrightarrow B^TB$.

So I would like to know whether the function $f\circ g:B\longrightarrow \mathrm{trace}((B^TB)^{-1})$ is convex and how we can prove that.

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Consider the case where $n=m=1$. Then your function is $s \mapsto 1/(s^2)$ which is not convex on the whole domain of its definition, i.e., $(-\infty,0)\cup(0,\infty)$. –  passerby51 Feb 12 '13 at 0:51
So definitely we cannot consider $f\circ g$ as a convex function. Thanks! –  user2987 Feb 12 '13 at 1:21