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You can use eigenvalue solver to find singular values of any matrix but not vica versa. Implementing eigenvalues needs complex numbers
by yaroslavvb 3mo ago
You can use eigenvalue solver to find singular values of any matrix but not vica versa. Implementing eigenvalues needs complex numbers
- xscott 3mo agoAnd 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 3mo ago> 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.