Suppose I have a matrix $H^{+} = (H^T H)^{-1} H^T$ where $H$ is a sparse matrix. Consider the case where only the sparsity pattern i.e. the zero elements of $H^{+}$ matrix is known, then would it be possible to infer the sparsity pattern or any other information of the $H$ matrix in either a deterministic or a probabilistic sense?
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$\begingroup$ Presumably $H$ has full column rank, since $(H^T H)^{-1}$ exists. If $H$ is a square matrix, $H^+ = H^{-1}$. But the inverse of an invertible sparse matrix is likely to be dense. $\endgroup$– Robert IsraelCommented May 31, 2018 at 6:18
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$\begingroup$ Since $H^+ H = I$, for each $i$ there is some $j$ such that $(H^+)_{ij} \ne 0$ and $H_{ji} \ne 0$. $\endgroup$– Robert IsraelCommented May 31, 2018 at 6:34
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