corpcor (version 1.5.6)

pseudoinverse: 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

$$M=UD{V}^{{}^{\prime }},$$

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 Moore-Penrose or generalized inverse is then obtained as $$iM=V{D}^{-1}{U}^{{}^{\prime }}$$

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)

Value

• A matrix (the pseudoinverse of m).

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.

See Also

solve, fast.svd

Examples

# load corpcor library
library("corpcor")

# 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) )