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Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want to represent it by a low rank approximation but respecting the symmetry and the zero diagonal in the output of that approximation. Like doing the eigenvalue decomposition by the spectral theorem $$ A=\sum_{i=1}^n \lambda_iv_iv_i^T $$ and then truncate this sum to get an approximation for $k\leq n$ (assuming $|\lambda_1|\geq\ldots\geq|\lambda_n|$). But if you do this for $A$ of course you get something like (with two eigenvalues) $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0.5 & 0.5\\ 2 & 0.5 & 0.5 },$$ which is not in the spirit of what I want.

Question: Is there a standard procedure/method for dimension reduction respecting these constraints?

EDIT 1: I realized that since you need to have a zero diagonal output as an approximation matrix, this implies full rank except if some of the entries are zero. So I guess the answer is in choosing which entries are set to zero and how the others are reweighted.

Thanks!

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