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Pseudoinverse

Usage

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

Basic examples

Find the pseudoinverse of an invertible matrix: In[1]:=m=(1 | 2 3 | 4); PseudoInverse[m] Out[1]={{-2, 1}, {3/2, -1/2}} The pseudoinverse is merely the inverse: In[2]:=%==Inverse[m] Out[2]=True Find the pseudoinverse of a singular matrix: In[1]:=m=(1 | 2 | 3 4 | 5 | 6 7 | 8 | 9); p=PseudoInverse[m] Out[1]={{-23/36, -1/6, 11/36}, {-1/18, 0, 1/18}, {19/36, 1/6, -7/36}} The determinant of m is zero, so it does not have a true inverse: In[2]:=Det[m] Out[2]=0 For a pseudoinverse, both m.p.m = m and p.m.p = p: In[3]:=m.p.m==m && p.m.p==p Out[3]=True However, in this particular case neither m.p nor p.m is an identity matrix: In[4]:={m.p//MatrixForm, p.m//MatrixForm} Out[4]={(5/6 | 1/3 | -1/6 1/3 | 1/3 | 1/3 -1/6 | 1/3 | 5/6), (5/6 | 1/3 | -1/6 1/3 | 1/3 | 1/3 -1/6 | 1/3 | 5/6)} Find the pseudoinverse of a rectangular matrix: In[1]:=m=(1 | 2 | 3 4 | 5 | 6); p=PseudoInverse[m] Out[1]={{-17/18, 4/9}, {-1/9, 1/9}, {13/18, -2/9}} In this particular case, m.p is an identity matrix: In[2]:=m.p Out[2]={{1, 0}, {0, 1}} However, p.m is not: In[3]:=p.m Out[3]={{5/6, 1/3, -1/6}, {1/3, 1/3, 1/3}, {-1/6, 1/3, 5/6}}

Option

Tolerance

Common option value

Tolerance | 0 | 10^(-15)

Attributes

NonThreadable | Protected

Relationships with other entities

Inverse | LeastSquares | Fit | SingularValueDecomposition | SingularValueList | DrazinInverse

Relationships with other entities

Typical ranks of usage in programs

1543rd most common (1 in 271000 symbols)

1159th most common (1 in 462000 symbols)

2253rd most common (1 in 57900 symbols)

History

introduced in Version 1 (June 1988) last modified in Version 5 (June 2003)

Timeline

Timeline

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