Singular Value Decomposition of a Matrix

## Arguments

- x
A \(m \times p\) numeric `matrix`

.

- rank
An `integer`

value specifying the maximal number of components
to be kept in the results.

## Value

A `list`

with the following elements:

`d`

A vector containing the singular values of `x`

, of length
`rank`

, sorted decreasingly.

`u`

A matrix whose columns contain the left singular vectors of
`x`

. Dimension `c(m, rank)`

.

`v`

A matrix whose columns contain the right singular vectors of
`x`

. Dimension `c(p, rank)`

.

## Note

In both PCA and PCA-cor whitening there is a sign-ambiguity in the
eigenvector matrices. In order to resolve the sign-ambiguity we use
eigenvector matrices with a positive diagonal. This has the effect to make
cross-correlations and cross-correlations positive diagonal for PCA.