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Somewhere in my reading, it seems that the following inequality holds for every symmetric positive definite matrix $A$, $$A\geq \big(\mathrm{tr}(A^{-1})\big)^{-1}I,$$ where $I$ is the identity matrix.

Is this true?

Thank you!

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Yes. By a change of orthonormal basis, you may assume that $A=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ and the inequality becomes $\operatorname{tr}(A^{-1})A\ge I$ or $\lambda_i\sum_{k=1}^n\frac{1}{\lambda_k}\ge1$ for every $i$.

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