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If $\Sigma$ is a real, symmetric, positive semidefinite matrix (a covariance matrix), then we can construct $M = \Sigma - \operatorname{diag}(\Sigma)$, where we essentially take the covariance matrix and set the diagonals to zero.

Is this matrix studied in any context, perhaps with a special name? For instance, one could ask whether there are relationships between eigendecomposition of $\Sigma$ and $M$. Obviously $M$ is still real, symmetric but no longer p.s.d.

The relationship between the graph Laplacian and adjacency matrix is somewhat similar, but the rows in $\Sigma$ don't have to sum to zero as they do for the Laplacian.

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