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Symmetric or nonsymmetric, that is the question

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Symmetric or nonsymmetric, that is the question

I have real, [ideally] symmetric matrices for which I need to compute the eigenvectors and eigenvalues. The concern that I have with choosing a method for determining the eigenstuff is that my matrices are only symmetric to within (roughly) the roundoff error propagation of all the computations leading up to them. Physically, they should be symmetric. I have looked at the effect of perturbing matrix A on the [possibly] perturbed eigenvalues of A. I note that the eigenvectors are more sensitive than the eigenvalues, because they depend on the eigenvalues and the distance twixt the eigenvalues. If the matrix has well-conditioned eigenvalues, which they should be for a [nearly] symmetric matrix, and are not too close together, the problem of computing the eigenvectors is likely well-conditioned, too. Not that I'm an expert. I can barely spell eigenwhatever.

At some point, one determines that a matrix is no longer really symmetric, but how do I make that decision? A few thoughts...the real problem is not round off error itself, but the relative magnitude of the error and how the actual values in the matrix relate to eps. Assuming that the real [possibly positive definite] matrix values are large compared to eps, and that I can say for each value

|1-(rounded/actual)| << 1,

then I am prolly OK. I'd like something more definitive than that if possible, but everywhere I look, I come up empty.