The Moore-Penrose inverse is a generalization of the regular inverse of a square, non-singular, symmetric matrix to other cases (rectangular, singular), yet retain similar properties to a regular inverse.
Usage
MoorePenrose(X, tol = sqrt(.Machine$double.eps))Examples
X <- matrix(rnorm(20), ncol=2)
# introduce a linear dependency in X[,3]
X <- cbind(X, 1.5*X[, 1] - pi*X[, 2])
Y <- MoorePenrose(X)
# demonstrate some properties of the M-P inverse
# X Y X = X
round(X %*% Y %*% X - X, 8)
#> [,1] [,2] [,3]
#> [1,] 0 0 0
#> [2,] 0 0 0
#> [3,] 0 0 0
#> [4,] 0 0 0
#> [5,] 0 0 0
#> [6,] 0 0 0
#> [7,] 0 0 0
#> [8,] 0 0 0
#> [9,] 0 0 0
#> [10,] 0 0 0
# Y X Y = Y
round(Y %*% X %*% Y - Y, 8)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 0 0 0 0 0 0 0 0 0 0
#> [2,] 0 0 0 0 0 0 0 0 0 0
#> [3,] 0 0 0 0 0 0 0 0 0 0
# X Y = t(X Y)
round(X %*% Y - t(X %*% Y), 8)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 0 0 0 0 0 0 0 0 0 0
#> [2,] 0 0 0 0 0 0 0 0 0 0
#> [3,] 0 0 0 0 0 0 0 0 0 0
#> [4,] 0 0 0 0 0 0 0 0 0 0
#> [5,] 0 0 0 0 0 0 0 0 0 0
#> [6,] 0 0 0 0 0 0 0 0 0 0
#> [7,] 0 0 0 0 0 0 0 0 0 0
#> [8,] 0 0 0 0 0 0 0 0 0 0
#> [9,] 0 0 0 0 0 0 0 0 0 0
#> [10,] 0 0 0 0 0 0 0 0 0 0
# Y X = t(Y X)
round(Y %*% X - t(Y %*% X), 8)
#> [,1] [,2] [,3]
#> [1,] 0 0 0
#> [2,] 0 0 0
#> [3,] 0 0 0