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I need to compute numerically the determinant of $M^T M$ where $M$ is a large (non-square) sparse matrix. Is there any sensible way to compute it? More generally, is there any mathematical property of determinants for expressions like $M^T M$?

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Assuming that M has more rows than columns (or else the determinant is 0), you could do a rank revealing qr decomposition; if M is rank deficient the determinant is zero, and otherwise we have

M'*M = R'*R

where R is square upper triangular and so det(R) is the product of its diagonal elements.

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