Strictly convex on the set of positive definite matrix

Question

I need some assistance in solving this;

Let $ A, B \in M_n$ be positive definite and let $\alpha \in (0,1)$. I need to show that $\alpha A^{-1} + (1-\alpha)B^{-1} \geq ((\alpha A+(1-\alpha)B)^{-1}$ with equality iff A=B. Thus the function $ f(t)=t^{-1}$ is strictly convex on the set of positive definite matrix

score 2 · Accepted Answer · edited Apr 13 '17 at 12:19

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By $\mathbf{X} \geq \mathbf{Y}$ do you mean $\mathbf{X}\succeq\mathbf{Y}$, which implies $\mathbf{X}-\mathbf{Y}$ is positive definite ? If so, then following post answers your question, Is inverse matrix convex?

edited Apr 13 '17 at 12:19

Community

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answered May 01 '14 at 22:25

Chetan Tonde

46

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Note that the accepted answer in that post is said to have a mistake. – Martin Argerami May 01 '14 at 22:45
I think the second answer by @achillehui is correct. – Chetan Tonde Jun 05 '14 at 14:41

Strictly convex on the set of positive definite matrix

1 Answers1