mixAK (version 5.8)

MatMPpinv: Moore-Penrose pseudoinverse of a squared matrix

Description

For a matrix \(\boldsymbol{A}\) its Moore-Penrose pseudoinverse is such a matrix \(\boldsymbol{A}^+\) which satisfies

(i) \(\boldsymbol{A}\boldsymbol{A}^+\boldsymbol{A} = \boldsymbol{A}\),

(ii) \(\boldsymbol{A}^+\boldsymbol{A}\boldsymbol{A}^+ = \boldsymbol{A}^+\),

(iii) \((\boldsymbol{A}\boldsymbol{A}^+)' = \boldsymbol{A}\boldsymbol{A}^+\),

(iv) \((\boldsymbol{A}^+\boldsymbol{A}) = \boldsymbol{A}^+\boldsymbol{A}\).

Computation is done using spectral decomposition. At this moment, it is implemented for symmetric matrices only.

Usage

MatMPpinv(A)

Value

Either a numeric vector or a matrix.

Arguments

A

either a numeric vector in which case inverse of each element of A is returned or a squared matrix.

Author

Arnošt Komárek

References

Golub, G. H. and Van Loan, C. F. (1996, Sec. 5.5). Matrix Computations. Third Edition. Baltimore: The Johns Hopkins University Press.

Examples

set.seed(770328)
A <- rWISHART(1, 5, diag(4))
Ainv <- MatMPpinv(A)

### Check the conditions
prec <- 13
round(A - A %*% Ainv %*% A, prec)
round(Ainv - Ainv %*% A %*% Ainv, prec)
round(A %*% Ainv - t(A %*% Ainv), prec)
round(Ainv %*% A - t(Ainv %*% A), prec)