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  (2)

m is a 4×3 matrix:

Compute using exact arithmetic:

Compute using machine arithmetic:

Compute using 24‐digit precision arithmetic:

Compute the pseudoinverse for a random complex 3×2 matrix:

Generalizations & Extensions  (1)

Compute symbolic result:

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]:

Introduced in 1988
 (1.0)
 |
Updated in 2003
 (5.0)