Moore-Penrose Pseudoinverse of a Real-valued Matrix

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Description

Computes the Moore-Penrose generalized inverse.

Usage

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MPinv(mat, tolerance = 100*.Machine$double.eps,
      rank = NULL, method = "svd")

Arguments

mat

a real matrix.

tolerance

A positive scalar which determines the tolerance for detecting zeroes among the singular values.

rank

Either NULL, in which case the rank of mat is determined numerically; or an integer specifying the rank of mat if it is known. No check is made on the validity of any non-NULL value.

method

Character, one of "svd", "chol". The specification method = "chol" is valid only for symmetric matrices.

Details

Real-valuedness is not checked, neither is symmetry when method = "chol".

Value

A matrix, with an additional attribute named "rank" containing the numerically determined rank of the matrix.

Author(s)

David Firth and Heather Turner

References

Harville, D. A. (1997). Matrix Algebra from a Statistician's Perspective. New York: Springer.

Courrieu, P. (2005). Fast computation of Moore-Penrose inverse matrices. Neural Information Processing 8, 25–29

See Also

ginv

Examples

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A <- matrix(c(1, 1, 0,
              1, 1, 0,
              2, 3, 4), 3, 3)
B <- MPinv(A)
A %*% B %*% A - A  # essentially zero
B %*% A %*% B - B  # essentially zero
attr(B, "rank")    # here 2

## demonstration that "svd" and "chol" deliver essentially the same
## results for symmetric matrices:
A <- crossprod(A)
MPinv(A) - MPinv(A, method = "chol") ##  (essentially zero) 

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