[lr1970]

> Implementing eigenvalues needs complex numbers

This is exactly why eigenvalues are less flexible than singular values. You can have a real valued matrix that does not have any eigenvalues in the field of real numbers, all eigenvalues are complex numbers. Examples: rotation matrixes in R^2 have no eigenvalues in R. Singular values, on the other hand, are always real (they are eigenvalues of the Hermitian matrix MM^*) and can be used the same way for the matrices over real (R) or complex (C) numbers, hence extra flexibility. Added bonus -- singular values are never negative.

Instead of "more flexible" a better statement would be -- eigenvalues convey more information than singular values.