MatMPpinv {mixAK}

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)

Arguments

A

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

Value

Either a numeric vector or a matrix.

Author(s)

Arnošt Komárek [e-mail]

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)