MatMPpinv function

Moore-Penrose pseudoinverse of a squared matrix

For a matrix $A$ its Moore-Penrose pseudoinverse is such a matrix $A[+]$ which satisfies

(i) c("$$", "$\n$", "$\tA*A[+]*A = A$") ,
(ii) c("$$", "$\n$", "$A[+]*A*A[+] = A[+]$") ,
(iii) c("$$", "$\n$", "$(A*A[+])' = A*A[+]$") ,
(iv) c("$$", "$\n$", "$(A[+]*A)' = A[+]*A$") .

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

MatMPpinv(A)

Arguments

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

Returns

Either a numeric vector or a matrix.

References

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

Author(s)

Arnošt Komárek arnost.komarek@mff.cuni.cz

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)