How to show that $f(X)=tr(X^{-1})$ is convex over the set of symmetric real positive definite matrices by showing that $g(t)=f(X+tV)$ is convex for any such $X$ and $V$? Clearly we need to show that $$g(t)\le (1-t)g(0)+tg(1)$$ but I'm stuck to find a simplified version of $tr((X+tV)^{-1})$.
Thank you for your help!
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Mostafa Ayaz
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Also, you might find this question useful too: Inverse of the sum of matrices – Miguel Oct 23 '18 at 20:54
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And Neumann series. – Miguel Oct 23 '18 at 21:02