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 (5)

Find the pseudoinverse of an invertible matrix:

The pseudoinverse is merely the inverse:

Find the pseudoinverse of a singular matrix:

The determinant of is zero, so it does not have a true inverse:

For a pseudoinverse, both and :

However, in this particular case neither nor is an identity matrix:

Find the pseudoinverse of a rectangular matrix:

In this particular case, is an identity matrix:

However, is not:

Find the pseudoinverse of a row matrix:

Find the pseudoinverse of a zero matrix:

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 returned 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 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 (8)

Equation Solving (4)

Solve the following system of equations using PseudoInverse:

Rewrite the system in matrix form:

The general solution is given by for an arbitrary vector :

Since the dropped out, the solution is unique, as can be verified using SolveValues:

Find all solutions of the following system of equations:

First, write the coefficient matrix , vector variable and constant vector :

Verify the rewrite:

The general solution is given by for an arbitrary vector :

Verify the solution:

Although there are three parameters, this solution represents a line:

This is because the null space of is one-dimensional:

Hence it is possible to reparameterize to eliminate two of the parameters:

This parameterization gives the answer in the same form as SolveValues:

Find the minimum Frobenius-norm solution to , with and as follows:

The minimum norm solution is :

Compute the Frobenius norm of :

As in the vector case, the general solution is given by , with now an arbitrary matrix:

The minimum occurs when all the are zero, confirming that is the minimum-norm solution:

In this case there is no solution to :

An approximate solution that minimizes the norm of is given by :

Compare to general minimization:

A more general solution is given by :

All these vectors minimize :

Although there are three parameters in , it represents a line:

This is because the null space of is one-dimensional:

Least Squares and Curve Fitting (4)

For the matrix and vector that follow, find a vector that minimizes :

One solution, in this case unique, is given by :

This result could also have been obtained using LeastSquares[m,b]:

Confirm the answer using Minimize:

For the matrices and that follow, find a matrix that minimizes $TemplateBox[{{{m, ., x}, -, b}}, Norm]_F$ :

One solution, in this case unique, is given by :

This result could also have been obtained using LeastSquares[m,b]:

Confirm the answer using Minimize:

PseudoInverse can be used to find a best-fit curve to data. Consider the following data:

Extract the and coordinates from the data:

Construct a design matrix, whose columns are and , for fitting to a line :

Get the coefficients and for a linear least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find the best-fit parabola to the following data:

Extract the and coordinates from the data:

Construct a design matrix, whose columns are , and , for fitting to a line :

Get the coefficients , and for a least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Properties & Relations (14)

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

PseudoInverse is involutive, :

PseudoInverse commutes with Transpose, i.e $(TemplateBox[{m}, Transpose])^((-1))=TemplateBox[{{(, {m, ^, {(, {(, {-, 1}, )}, )}}, )}}, Transpose]$ :

It also commutes with Conjugate, $(TemplateBox[{m}, Conjugate])^((-1))=TemplateBox[{{(, {m, ^, {(, {(, {-, 1}, )}, )}}, )}}, Conjugate]$ :

Hence it commutes with ConjugateTranspose, $(TemplateBox[{m}, ConjugateTranspose])^((-1))=TemplateBox[{{(, {m, ^, {(, {(, {-, 1}, )}, )}}, )}}, ConjugateTranspose]$ :

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

If MatrixRank[m] equals the number of columns of , then $m^((-1))=TemplateBox[{{(, {TemplateBox[{m}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], ., m}, )}}, Inverse].TemplateBox[{m}, ConjugateTranspose]$ :

In particular, PseudoInverse[m] is a left-inverse of m:

If MatrixRank[m] equals the number of rows of , then $m^((-1))=TemplateBox[{m}, ConjugateTranspose].TemplateBox[{{(, {m, ., TemplateBox[{m}, ConjugateTranspose]}, )}}, Inverse]$ :