Generalized Inverse of a Matrix

Ginv returns an arbitrary generalized inverse of the matrix A, using gaussianElimination.

Usage

Ginv(A, tol = sqrt(.Machine$double.eps), verbose = FALSE, fractions = FALSE)

Arguments

A: numerical matrix
tol: tolerance for checking for 0 pivot
verbose: logical; if TRUE, print intermediate steps
fractions: logical; if TRUE, try to express non-integers as rational numbers, using the fractions function; if you require greater accuracy, you can set the cycles (default 10) and/or max.denominator (default 2000) arguments to fractions as a global option, e.g., options(fractions=list(cycles=100, max.denominator=10^4)).

Value

the generalized inverse of A, expressed as fractions if fractions=TRUE, or rounded

Details

A generalized inverse is a matrix \(\mathbf{A}^-\) satisfying \(\mathbf{A A^- A} = \mathbf{A}\).

The purpose of this function is mainly to show how the generalized inverse can be computed using Gaussian elimination.

Author

John Fox

Examples

A <- matrix(c(1,2,3,4,5,6,7,8,10), 3, 3) # a nonsingular matrix
A
#>      [,1] [,2] [,3]
#> [1,]    1    4    7
#> [2,]    2    5    8
#> [3,]    3    6   10
Ginv(A, fractions=TRUE)  # a generalized inverse of A = inverse of A
#>      [,1] [,2] [,3]
#> [1,] -2/3 -2/3    1
#> [2,] -4/3 11/3   -2
#> [3,]    1   -2    1
round(Ginv(A) %*% A, 6)  # check
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1

B <- matrix(1:9, 3, 3) # a singular matrix
B
#>      [,1] [,2] [,3]
#> [1,]    1    4    7
#> [2,]    2    5    8
#> [3,]    3    6    9
Ginv(B, fractions=TRUE)  # a generalized inverse of B
#>      [,1]  [,2]  [,3] 
#> [1,]  -3/4     0  7/12
#> [2,]     0     0     0
#> [3,]   1/4     0 -1/12
B %*% Ginv(B) %*% B   # check
#>      [,1] [,2] [,3]
#> [1,]    1    4    7
#> [2,]    2    5    8
#> [3,]    3    6    9