Hello all,

I come across the following problem.

Is it true that for a **positive definite** matrix $X^{n\times n}$, the following holds

$\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$,

where $\text{trace}(\;\cdot\;)$ and $\text{diag}(\;\cdot\;)$ are the matrix trace and diagonal operators, respectively.

Remark 1: Note that in the above, there is equality when $X$ is diagonal.

Remark 2: By using the SVD of the matrix $X$, or by using the Hadamard inequality, the above inequality is equivalent to

$\sum\limits_{i=1}^n\frac{1}{\lambda_i}-\sum\limits_{i=1}^n\frac{1}{x_{ii}}\geq0$,

where $\lambda_i$ and $x_{ii}$ are the $i$th eigenvalue of the matrix $X$, and the $i$th diagonal element of the matrix $X$, respectively.