PseudoInverse
✖
PseudoInverse
Details and Options
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- 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
open allclose allBasic Examples (5)Summary of the most common use cases
Find the pseudoinverse of an invertible matrix:
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https://wolfram.com/xid/0rs0l2k2a-htoxr7
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The pseudoinverse is merely the inverse:
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https://wolfram.com/xid/0rs0l2k2a-tyzi8r
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Find the pseudoinverse of a singular matrix:
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https://wolfram.com/xid/0rs0l2k2a-dn2wo0
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The determinant of is zero, so it does not have a true inverse:
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https://wolfram.com/xid/0rs0l2k2a-8ssntl
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For a pseudoinverse, both and
:
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https://wolfram.com/xid/0rs0l2k2a-rvdqhc
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However, in this particular case neither nor
is an identity matrix:
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https://wolfram.com/xid/0rs0l2k2a-tuk8ac
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Find the pseudoinverse of a rectangular matrix:
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https://wolfram.com/xid/0rs0l2k2a-mrekcd
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In this particular case, is an identity matrix:
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https://wolfram.com/xid/0rs0l2k2a-6irvfh
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https://wolfram.com/xid/0rs0l2k2a-zcejrl
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Find the pseudoinverse of a row matrix:
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https://wolfram.com/xid/0rs0l2k2a-nhxt3
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Find the pseudoinverse of a zero matrix:
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https://wolfram.com/xid/0rs0l2k2a-cffd0i
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Scope (10)Survey of the scope of standard use cases
Basic Uses (6)
Find the pseudoinverse of a machine-precision matrix:
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https://wolfram.com/xid/0rs0l2k2a-gzis3e
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Pseudoinverse of a complex matrix:
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https://wolfram.com/xid/0rs0l2k2a-bjja1q
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Pseudoinverse of an exact matrix:
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https://wolfram.com/xid/0rs0l2k2a-piv3b5
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Pseudoinverse of an arbitrary-precision matrix:
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https://wolfram.com/xid/0rs0l2k2a-p8sjhl
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Compute a symbolic pseudoinverse:
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https://wolfram.com/xid/0rs0l2k2a-gtn59g
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The inversion of large machine-precision matrices is efficient:
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https://wolfram.com/xid/0rs0l2k2a-dkq7nk
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https://wolfram.com/xid/0rs0l2k2a-lx8juz
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Special Matrices (4)
The pseudoinverse of a sparse matrix is returned as a normal matrix:
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https://wolfram.com/xid/0rs0l2k2a-ocj3kf
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https://wolfram.com/xid/0rs0l2k2a-fm3xwv
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https://wolfram.com/xid/0rs0l2k2a-eordwx
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When possible, the pseudoinverse of a structured matrix is returned as another structured matrix:
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https://wolfram.com/xid/0rs0l2k2a-ebpduc
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https://wolfram.com/xid/0rs0l2k2a-dhkxvx
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https://wolfram.com/xid/0rs0l2k2a-wdjmwi
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https://wolfram.com/xid/0rs0l2k2a-j45yx1
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IdentityMatrix[n] is its own pseudoinverse:
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https://wolfram.com/xid/0rs0l2k2a-8vb729
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The pseudoinverse of IdentityMatrix[{m,n}] is a transposition:
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https://wolfram.com/xid/0rs0l2k2a-pflv9c
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https://wolfram.com/xid/0rs0l2k2a-89ddq0
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Compute the pseudoinverse for HilbertMatrix:
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https://wolfram.com/xid/0rs0l2k2a-q1l839
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Options (1)Common values & functionality for each option
Tolerance (1)
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https://wolfram.com/xid/0rs0l2k2a-g33jlv
Some singular values are below the default tolerance for machine precision:
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https://wolfram.com/xid/0rs0l2k2a-hkuhvt
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Compute the pseudoinverse with the default tolerance:
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https://wolfram.com/xid/0rs0l2k2a-b0qgc0
It is not a true inverse since some singular values were considered to be effectively zero:
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https://wolfram.com/xid/0rs0l2k2a-cgimnl
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Compute the pseudoinverse with no tolerance:
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https://wolfram.com/xid/0rs0l2k2a-h80yg5
Even though no singular values were considered zero, it is worse due to numerical error:
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https://wolfram.com/xid/0rs0l2k2a-5olrz
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Applications (8)Sample problems that can be solved with this function
Equation Solving (4)
Solve the following system of equations using PseudoInverse:
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https://wolfram.com/xid/0rs0l2k2a-upikkb
Rewrite the system in matrix form:
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https://wolfram.com/xid/0rs0l2k2a-zs7ify
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The general solution is given by for an arbitrary vector
:
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https://wolfram.com/xid/0rs0l2k2a-qytjf
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Since the dropped out, the solution is unique, as can be verified using SolveValues:
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https://wolfram.com/xid/0rs0l2k2a-tqujfz
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Find all solutions of the following system of equations:
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https://wolfram.com/xid/0rs0l2k2a-1c6b95
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First, write the coefficient matrix , vector variable
and constant vector
:
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https://wolfram.com/xid/0rs0l2k2a-vokjmu
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https://wolfram.com/xid/0rs0l2k2a-se2owl
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The general solution is given by for an arbitrary vector
:
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https://wolfram.com/xid/0rs0l2k2a-m3h2nw
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https://wolfram.com/xid/0rs0l2k2a-kyqgwy
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Although there are three parameters, this solution represents a line:
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https://wolfram.com/xid/0rs0l2k2a-1yawfd
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This is because the null space of is one-dimensional:
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https://wolfram.com/xid/0rs0l2k2a-ihejh6
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Hence it is possible to reparameterize to eliminate two of the parameters:
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https://wolfram.com/xid/0rs0l2k2a-dj3tes
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This parameterization gives the answer in the same form as SolveValues:
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https://wolfram.com/xid/0rs0l2k2a-14h576
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Find the minimum Frobenius-norm solution to , with
and
as follows:
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https://wolfram.com/xid/0rs0l2k2a-ziuqcu
The minimum norm solution is :
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https://wolfram.com/xid/0rs0l2k2a-nmbplo
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Compute the Frobenius norm of :
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https://wolfram.com/xid/0rs0l2k2a-hoiqzn
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As in the vector case, the general solution is given by , with
now an arbitrary matrix:

https://wolfram.com/xid/0rs0l2k2a-q66wu2


https://wolfram.com/xid/0rs0l2k2a-vk1bir

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

https://wolfram.com/xid/0rs0l2k2a-eb2arl

In this case there is no solution to :

https://wolfram.com/xid/0rs0l2k2a-cmu0c9

https://wolfram.com/xid/0rs0l2k2a-g81vfi

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

https://wolfram.com/xid/0rs0l2k2a-o6nyhb
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https://wolfram.com/xid/0rs0l2k2a-g2o5mo
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Compare to general minimization:

https://wolfram.com/xid/0rs0l2k2a-g5rxbt
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A more general solution is given by :

https://wolfram.com/xid/0rs0l2k2a-5uz3ix


https://wolfram.com/xid/0rs0l2k2a-qusxb6

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

https://wolfram.com/xid/0rs0l2k2a-107pgz

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

https://wolfram.com/xid/0rs0l2k2a-p5kqmy

Least Squares and Curve Fitting (4)
For the matrix and vector
that follow, find a vector
that minimizes
:

https://wolfram.com/xid/0rs0l2k2a-e4enjw
One solution, in this case unique, is given by :

https://wolfram.com/xid/0rs0l2k2a-s7lr4r

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

https://wolfram.com/xid/0rs0l2k2a-2cpasl

Confirm the answer using Minimize:

https://wolfram.com/xid/0rs0l2k2a-w70jbo

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

https://wolfram.com/xid/0rs0l2k2a-pqffuw
One solution, in this case unique, is given by :

https://wolfram.com/xid/0rs0l2k2a-ivgyxa

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

https://wolfram.com/xid/0rs0l2k2a-zp70nm

Confirm the answer using Minimize:

https://wolfram.com/xid/0rs0l2k2a-4kcdw8

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

https://wolfram.com/xid/0rs0l2k2a-v42zji

Extract the and
coordinates from the data:

https://wolfram.com/xid/0rs0l2k2a-i4v6jo
Construct a design matrix, whose columns are and
, for fitting to a line
:

https://wolfram.com/xid/0rs0l2k2a-pkve26
Get the coefficients and
for a linear least‐squares fit:

https://wolfram.com/xid/0rs0l2k2a-7p26fs

Verify the coefficients using Fit:

https://wolfram.com/xid/0rs0l2k2a-480xbe
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Plot the best-fit curve along with the data:

https://wolfram.com/xid/0rs0l2k2a-wzqna8
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Find the best-fit parabola to the following data:

https://wolfram.com/xid/0rs0l2k2a-x7pawr

Extract the and
coordinates from the data:

https://wolfram.com/xid/0rs0l2k2a-w5dwto
Construct a design matrix, whose columns are ,
and
, for fitting to a line
:

https://wolfram.com/xid/0rs0l2k2a-bmqc8e
Get the coefficients ,
and
for a least‐squares fit:

https://wolfram.com/xid/0rs0l2k2a-z5bw8q
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Verify the coefficients using Fit:

https://wolfram.com/xid/0rs0l2k2a-pp5xfy

Plot the best-fit curve along with the data:

https://wolfram.com/xid/0rs0l2k2a-04d0qo
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Properties & Relations (14)Properties of the function, and connections to other functions
For a nonsingular matrix, the pseudoinverse is the same as the inverse:

https://wolfram.com/xid/0rs0l2k2a-g2efuw

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https://wolfram.com/xid/0rs0l2k2a-hq2ojv
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PseudoInverse is involutive, :

https://wolfram.com/xid/0rs0l2k2a-ci246w

https://wolfram.com/xid/0rs0l2k2a-zwdjpl

PseudoInverse commutes with Transpose, i.e :

https://wolfram.com/xid/0rs0l2k2a-r5artr
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It also commutes with Conjugate, :

https://wolfram.com/xid/0rs0l2k2a-s204v

Hence it commutes with ConjugateTranspose, :

https://wolfram.com/xid/0rs0l2k2a-ee2zf9
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PseudoInverse satisfies the Moore–Penrose equations [more info]:

https://wolfram.com/xid/0rs0l2k2a-p2fhsi

https://wolfram.com/xid/0rs0l2k2a-bs1pa5

https://wolfram.com/xid/0rs0l2k2a-bsvl0m

If MatrixRank[m] equals the number of columns of , then
:

https://wolfram.com/xid/0rs0l2k2a-0x2fwg
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
https://wolfram.com/xid/0rs0l2k2a-sgmkky
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In particular, PseudoInverse[m] is a left-inverse of m:

https://wolfram.com/xid/0rs0l2k2a-n8bjnc
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If MatrixRank[m] equals the number of rows of , then
:

https://wolfram.com/xid/0rs0l2k2a-s17ijx


https://wolfram.com/xid/0rs0l2k2a-c2g8j6
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In particular, PseudoInverse[m] is a right-inverse of m:

https://wolfram.com/xid/0rs0l2k2a-t4cimv
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For a diagonal matrix d, PseudoInverse[d] is the transpose with nonzero elements inverted:

https://wolfram.com/xid/0rs0l2k2a-d17ssz

If has the singular value decomposition
, then
:

https://wolfram.com/xid/0rs0l2k2a-v34myz

https://wolfram.com/xid/0rs0l2k2a-s69vp9
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https://wolfram.com/xid/0rs0l2k2a-mdilnn

If a is an matrix and MatrixRank[a]==m, QRDecomposition will give the pseudoinverse:

https://wolfram.com/xid/0rs0l2k2a-zpyy8q


https://wolfram.com/xid/0rs0l2k2a-x6p50r
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A normal matrix commutes with its pseudoinverse:
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https://wolfram.com/xid/0rs0l2k2a-n714iw
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https://wolfram.com/xid/0rs0l2k2a-nzl1fk
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https://wolfram.com/xid/0rs0l2k2a-cvfe76
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PseudoInverse[m] can be computed as , where
denotes DrazinInverse:
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https://wolfram.com/xid/0rs0l2k2a-cceowv
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https://wolfram.com/xid/0rs0l2k2a-ei7hch
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LeastSquares and PseudoInverse can both be used to solve the least-squares problem:
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https://wolfram.com/xid/0rs0l2k2a-h3z3e9
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https://wolfram.com/xid/0rs0l2k2a-e5dgz8
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gives the minimum norm
that minimizes the residual
:
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https://wolfram.com/xid/0rs0l2k2a-lbfrc5
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https://wolfram.com/xid/0rs0l2k2a-884c7
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Verify that minimizes the residual:
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https://wolfram.com/xid/0rs0l2k2a-9nhura
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Adding any vector in the NullSpace of to
will leave the residual unchanged:
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https://wolfram.com/xid/0rs0l2k2a-7iufp
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https://wolfram.com/xid/0rs0l2k2a-c5eqo0
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The minimum norm occurs at
, i.e when
:
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https://wolfram.com/xid/0rs0l2k2a-b2f6yi
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For a vector and a matrix
with empty nullspace,
equals ArgMin[Norm[m.x-b],x]:
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https://wolfram.com/xid/0rs0l2k2a-c6zrbw
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https://wolfram.com/xid/0rs0l2k2a-9lyuvk
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https://wolfram.com/xid/0rs0l2k2a-fvs5bz
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Wolfram Research (1988), PseudoInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/PseudoInverse.html (updated 2003).
Text
Wolfram Research (1988), PseudoInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/PseudoInverse.html (updated 2003).
Wolfram Research (1988), PseudoInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/PseudoInverse.html (updated 2003).
CMS
Wolfram Language. 1988. "PseudoInverse." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PseudoInverse.html.
Wolfram Language. 1988. "PseudoInverse." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PseudoInverse.html.
APA
Wolfram Language. (1988). PseudoInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PseudoInverse.html
Wolfram Language. (1988). PseudoInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PseudoInverse.html
BibTeX
@misc{reference.wolfram_2025_pseudoinverse, author="Wolfram Research", title="{PseudoInverse}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PseudoInverse.html}", note=[Accessed: 24-February-2025
]}
BibLaTeX
@online{reference.wolfram_2025_pseudoinverse, organization={Wolfram Research}, title={PseudoInverse}, year={2003}, url={https://reference.wolfram.com/language/ref/PseudoInverse.html}, note=[Accessed: 24-February-2025
]}