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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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