[yaroslavvb]

Eigenvalue are more flexible than singular values -- If you have procedure for computing eigenvalues of C, squared singular values of X are eigenvalues of C=XX'. Left and write singular vectors of X and eigenvectors of XX' and X'X. For rotation rotation matrix, singular values are all 1. Meanwhile eigenvalues tell you angles of rotation https://math.stackexchange.com/questions/4874616/obtaining-a...

[xscott]

That's like saying spoons are more flexible than forks because you have soup (rotation matrices). Spoons and forks both work for rice (pos def matrices), and you'll want a fork for noodles (rectangular matrices).

[yaroslavvb]

You can use eigenvalue solver to find singular values of any matrix but not vica versa. Implementing eigenvalues needs complex numbers

[xscott]

And you can use a spoon to chop spaghetti into pieces. Say I have a 10,000 by 100 matrix (1M elements), squaring that to get a 10,000 x 10,000 (100M elements) matrix to get the singular vectors is probably a bad idea. That squares the condition number too.

The SVD staying in Reals when you have Real data is a nice feature.

[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.