Suppose we have a symmetric positive definite random matrix $\mathbf{A}$ that has a diagonal matrix expectation, $$ \mathbf{E}(\mathbf{A}) = \text{diag}(\mathbf{x}), $$ [like e.g. the Wishart distribution with diagonal scale matrix, $\mathcal{W}(\text{diag}(\mathbf{x}),n)$].

Is it true that $$ \mathbf{E}(\mathbf{A}^{-k}), $$ where $k$ is an arbitrary positive integer is also a diagonal matrix?

Edit: For the Wishart this is true. I am interested in the general statement.



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