svd — Singular Value Decomposition

$\begin{rail} SVD : 'svd' '(' Matrix (',' String ) ')' ; \end{rail}$
svd calculates the singular values, and different forms of the left and right singular vectors. The result is returned as three elements in a list. If we are performing singular value decomposition of A, then

A = UIσV^T

where U is the right singular vector, returned in u, σ is a vector of singular values, returned in sigma, and V^T is the left singular value, returned in vt. Full singular values area always returned. However the form of the left and right singular vectors is dependant on the value of the optional string argument:

"A" or "a": The full U, and V^T are returned
"S" or "s": A minimal version of U, and V^T are returned. This is the default.
"N" or "n": U and V^T are not computed, empty u and vt elements are returned.

Full singular vectors means that the null-space is represented in the singular values. This is the slowest option. Minimal singular vectors means that sufficient U and V^T are returned to reproduce A. In general, for an argument of dimensions rxc, then if r≥c,

Format	U	σ	V^T
Full	rxr	1xc	cxc
Minimal	rxc	1xc	cxc
Nil	0x0	1xc	0x0

For r≤c, then

Format	U	σ	V^T
Full	rxr	1xr	cxc
Minimal	rxr	1xr	rxc
Nil	0x0	1xr	0x0

Subsections

- - Example