7.4. Singular Value Decomposition

SVD (Singular Value Decomposition) stands for splitting a matrix \(A\) into a product \(A = U S V^H\) where \(U\) and \(V\) are unitary matrices and \(S\) is a diagonal matrix consisting of singular values on its main diagonal arranged in non-increasing order where all the singular values are non-negative.

Computing the singular value decomposition of a matrix:

> A <- matrix(c(1, 2, -1, 3, -6, 9, -1, 2, 1), nrow=3)
> svd(A)
$d
[1] 11.355933  2.475195  1.707690

$u
           [,1]       [,2]       [,3]
[1,]  0.2526715 -0.1073216 -0.9615816
[2,] -0.5565826  0.7968092 -0.2351827
[3,]  0.7914373  0.5946235  0.1415976

$v
            [,1]      [,2]       [,3]
[1,] -0.14546854 0.3602437 -0.9214464
[2,]  0.98806904 0.1005140 -0.1166899
[3,] -0.05058143 0.9274273  0.3705672

Largest singular value:

svd(A)$d[1]

Smallest singular value:

> tail(svd(A)$d, n=1)
[1] 1.70769

Absolute value of determinant as product of singular values:

> det(A)
[1] -48
> prod(svd(A)$d)
[1] 48