Are "singular values" and "eigenvalues" the same thing?



Are “singular values” and “eigenvalues” the same thing?


In a sense singular values are a generalization of eigenvalues in that eigenvalues are only defined for square matrices where, by contrast, you can find singular values for any matrix.

But even when your matrix is square, the eigenvalues will only be the same as the singular values when your matrix is symmetric and all of its eigenvalues are nonnegative. If some of them are negative then your singular values will just be some different numbers associated with the matrix.

Intuitively, eigenvalues tell you how much vectors are stretched/ contracted in the directions where they are only stretched/ contracted (in the directions of the eigenvectors).

Singular values, on the other hand, tell you something like the amount in scaling in the orthogonal frame where the scaling is maximized. IDK if that makes sense, so let’s take an example: if you apply an invertible linear transformation to the unit circle in R2, then the two singular values of the linear transformation (/ the matrix representing it) are the lengths of the semimajor and semiminor axes.

So they are somewhat similar (they are give information about scaling) but are not directly related.