BUILT-IN MATHEMATICA SYMBOL

PseudoInverse

PseudoInverse[m]
finds the pseudoinverse of a rectangular matrix.

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 , where p is Precision[m].

For non-singular square matrices M, the pseudoinverse is equivalent to the standard inverse.

A matrix has a pseudoinverse even if it is singular:

Out[1]=

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:

Compute symbolic result:

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:

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:

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

For

,

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 []: