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In the literature I have on disposal it is stated that singular values are non-negative values, and that, for a symmetric matrix $A$, the SVD and EVD coincide. This would mean that singular values of $A$ are the eigenvalues of $A$, but the eigenvalues of $A$ can be negative, regardless of $A$ being symmetric.

So, I wonder if the choice of singular values being exclusively positive is some kind of convention? If so, how degenerate that is given the above observation the equivalence of SVD and EVD for symmetric matrices?

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    @J.D. not only positive, nonnegative! (i.e., $\mathbf A$ is symmetric positive semidefinite)2012-07-14

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You can factor a (not necessarily square) matrix as orthogonal times diagonal times orthogonal, and the diagonal entries need not all be non-negative. But multiplying a row or a column of an orthogonal matrix by $-1$ still gives an orthogonal matrix, and you can do that and change a minus to a plus in the diagonal matrix. In that way, the two orthogonal matrices can be chosen so that the diagonal entries in that matrix are all non-negative. Those are what are taken to be the singular values.

It's a convention to define it that way. But I suspect there are theorems that say that's the only way to define it that makes it have specified nice properties, and those theorems would not be mere conventions.

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    @user506901, forming the cross-product matrix is in fact a bad idea (in inexact arithmetic), since the singular values are not as accurately determined in this case.2012-07-31