PseudoInverse

PseudoInverse[m]

finds the pseudoinverse of a rectangular matrix.

Details and Options

  • PseudoInverse works on both symbolic and numerical matrices.
  • For a square matrix, PseudoInverse gives the Moore–Penrose inverse.
  • For numerical matrices, PseudoInverse is based on SingularValueDecomposition.
  • PseudoInverse[m,Tolerance->t] specifies that singular values smaller than t times the maximum singular value should be dropped.
  • With the default setting Tolerance->Automatic, singular values are dropped when they are less than 100 times 10-p, where p is Precision[m].
  • For non‐singular square matrices M, the pseudoinverse M(-1) is equivalent to the standard inverse.

Examples

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Basic Examples  (1)

A matrix has a pseudoinverse even if it is singular:

Scope  (10)

Basic Uses  (6)

Find the pseudoinverse of a machine-precision matrix:

Pseudoinverse of a complex matrix:

Pseudoinverse of an exact matrix:

Pseudoinverse of an arbitrary-precision matrix:

Compute a symbolic pseudoinverse:

The inversion of large machine-precision matrices is efficient:

Special Matrices  (4)

The pseudoinverse of a sparse matrix is return as a normal matrix:

Format the result:

When possible, the pseudoinverse of a structured matrix is returned as another structured matrix:

This is not always possible:

IdentityMatrix[n] is its own pseudoinverse:

The pseudoinverse of IdentityMatrix[{m,n}] is a transposition:

Compute the the pseudoinverse for HilbertMatrix:

Options  (1)

Tolerance  (1)

m is a 16×16 Hilbert matrix:

Some singular values are below the default tolerance for machine precision:

Compute the pseudoinverse with the default tolerance:

It is not a true inverse since some singular values were considered to be effectively zero:

Compute the pseudoinverse with no tolerance:

Even though no singular values were considered zero, it is worse due to numerical error:

Applications  (1)

Here is some data:

Construct a design matrix for fitting to a line:

Get the coefficients for a linear least‐squares fit:

This is the same as the result given by Fit:

Properties & Relations  (3)

For a nonsingular matrix, the pseudoinverse is the same as the inverse:

For p = PseudoInverse[m], x = p.b gives the minimum norm x that minimizes :

Adding any vector in the NullSpace of m will leave the residual unchanged:

The minimum is at :

PseudoInverse satisfies the Moore–Penrose equations [more info]:

Wolfram Research (1988), PseudoInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/PseudoInverse.html (updated 2003).

Text

Wolfram Research (1988), PseudoInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/PseudoInverse.html (updated 2003).

BibTeX

@misc{reference.wolfram_2021_pseudoinverse, author="Wolfram Research", title="{PseudoInverse}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PseudoInverse.html}", note=[Accessed: 23-July-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_pseudoinverse, organization={Wolfram Research}, title={PseudoInverse}, year={2003}, url={https://reference.wolfram.com/language/ref/PseudoInverse.html}, note=[Accessed: 23-July-2021 ]}

CMS

Wolfram Language. 1988. "PseudoInverse." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PseudoInverse.html.

APA

Wolfram Language. (1988). PseudoInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PseudoInverse.html