Our definition of positive definiteness in isposdef is symmetric (Hermitian) and “can be Cholesky factorized” which is pretty close to having positive eigenvalues but not as robust. We have had some discussions about using a tolerance for the symmetry check but we concluded that it was better to make the users enforce symmetry via the Symmetric wrapper instead of coming up with a possibly arbitrary tolerance for symmetry. Hence, as long as you keep your matrix wrapped in the Symmetric type, a small permutation shouldn’t change the result of isposdef unless the matrix is almost singular.
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