pseudoinverse {GeneTS} R Documentation

## Pseudoinverse of a Matrix

### Description

The standard definition for the inverse of a matrix fails if the matrix is not square or singular. However, one can generalize the inverse using singular value decomposition. Any rectangular real matrix M can be decomposed as

M = U D V',

where U and V are orthogonal, V' means V transposed, and D is a diagonal matrix containing only the positive singular values (as determined by `tol`, see also `fast.svd`).

The pseudoinverse, also known as Moore­Penrose or generalized inverse is then obtained as

iM = V D^(-1) U' .

### Usage

```pseudoinverse(m, tol)
```

### Arguments

 `m` matrix `tol` tolerance - singular values larger than tol are considered non-zero (default value: `tol = max(dim(m))*max(D)*.Machine\$double.eps`)

### Details

The pseudoinverse has the property that the sum of the squares of all the entries in `iM %*% M - I`, where I is an appropriate identity matrix, is minimized. For non-singular matrices the pseudoinverse is equivalent to the standard inverse.

### Value

A matrix (the pseudoinverse of m).

### Author(s)

Korbinian Strimmer (http://www.stat.uni-muenchen.de/~strimmer/).

`solve`, `fast.svd`

### Examples

```# load GeneTS library
library(GeneTS)

# a singular matrix
m <- rbind(
c(1,2),
c(1,2)
)

# not possible to invert exactly
try(solve(m))

# pseudoinverse
p <- pseudoinverse(m)
p

# characteristics of the pseudoinverse
zapsmall( m %*% p %*% m )  ==  zapsmall( m )
zapsmall( p %*% m %*% p )  ==  zapsmall( p )
zapsmall( p %*% m )  ==  zapsmall( t(p %*% m ) )
zapsmall( m %*% p )  ==  zapsmall( t(m %*% p ) )

# example with an invertable matrix
m2 <- rbind(
c(1,1),
c(1,0)
)
zapsmall( solve(m2) ) == zapsmall( pseudoinverse(m2) )
```

[Package GeneTS version 2.3 Index]