Pinv: Compute the Moore-Penrose Pseudo Inverse of a matrix and...

In psych: Procedures for Psychological, Psychometric, and Personality Research

Compute the Moore-Penrose Pseudo Inverse of a matrix and other tools

Description

Given a matrix of less than full rank, the conventional inverse function will fail. The pseudoinverse or generalized inverse resolves this problem by using just the postive values of the singular value decomposition d matrix. This is an adaptation of the ginv function from MASS and the pinv function from pracma. Includes several other simple matrix tools such as matSqrt and invMatSqrt which can be used for “whitening" a matrix.

Usage

Pinv(R, tol = sqrt(.Machine$double.eps))
matSqrt(x) #take the sqrt of a matrix
invMatSqrt(x) #find the sqrt of the inverse matrix

Arguments

R	A correlation or covariance matrix to analyze
x	A correlation or covariance matrix to analyze
tol	A very small number. Reject values with eigen values less than tolerance

Details

The singular value decomposition of a matrix X is UdV where for full rank matrices, d is the vector of eigen values and U and V are the matrices of eigen vectors. The inverse is just U(1/d) U. If the matrix is less than full rank, many of the d values are effectively zero (at the limit of computational accuracy.) Thus, to solve matrix equations with matrices of less than full rank (e.g. the schmid Schmid-Leiman solution), we need to find the generalized inverse.

matSqrt and invMatSqrt will find the square root of a matrix or the square root of the Pinv (Penrose pseudo inverse) of a matrix.

invMatSqrt can be used to find the “whitening" matrix of a correlation or covariances such that for a correlation matrix, R, W= invMatSqrt(R):

W = R^{-.5}

and

W R W' = I.

Value

The generalized inverse, or the sqrt of the generalized inverse, or the sqrt of the Penrose inverse.

Note

Adapted from the ginv function in MASS and the pinv function in pracma. Installed here to avoid loading those packages.

Author(s)

William Revelle

References

Agnan Kessy, Alex Lewin and Korbinian Strimmer (2018) Optimal Whitening and Decorrelation, The American Statistician, 72:4, 309-314.

Marco Del Giudice (2024) Data matrix disattenuation: a simple, effective method for correcting measurement error in multivariate datasets. Preprint: PsyArXiv

Venables, W. N. and Ripley, B. D. (1999) Modern Applied Statistics with S-PLUS. Third Edition. Springer. p.100.Optimal

See Also

schmid, faCor

Examples

round(Pinv(Thurstone) %*% Thurstone,2)  #an identity matrix


W <- invMatSqrt(Thurstone) #find the whitening of Thurstone
round(W %*% Thurstone %*% W,2)

psych documentation built on Sept. 30, 2026, 5:08 p.m.