# PseudoInverse

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 MoorePenrose 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 , singular values are dropped when they are less than 100 times 10-p, where p is Precision[m].
• For nonsingular 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:

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 leastsquares 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 :