Mathematics Stack Exchange
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Questions tagged [pseudoinverse]

The operator which best approximates a solution to a linear system with a singular (non-invertible) matrix.: e.g., the Moore-Penrose pseudoinverse. Use when a question concerns a matrix that is probably singular.

This tag applies to, but is not restricted to:

Moore-Penrose Pseudoinverse

(also "Generalized Inverse") For linear operators and matrices that are not invertible there still exists a unique Moore-Penrose Pseudoinverse $A^+$ which fulfils the following conditions:

  • $A A^+A = A$ ($AA^+$ need not be the general identity matrix, but it maps all column vectors of $A$ to themselves);
  • $A^+A A^+ = A^+$ ($A^+$ is a weak inverse for the multiplicative semigroup);
  • $(AA^+)^* = AA^+$ ($AA^+$ is Hermitian); and
  • $(A^+A)^* = A^+A$ ($A^+A$ is also Hermitian).

Use it: when the Matrix/Operator involved is (probably) singular
Don't use it: when the Matrix/Operator is definitely invertible or its state is unknown

551 questions
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Blockwise Moore-Penrose pseudoinverse?

There exists a convenient formula for computing the inverse of a block matrix consisting of 4 matrices $\mathbf{A, B, C, D}$ $ \begin{bmatrix}\mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D}\end{bmatrix} ^{-1}$ the inverse can be written as a…
Wouter
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Meaning of the inverse of the Laplacian matrix

Given an undirected graph $G = (V,E)$, let $\bf A$ and $\cal L_{\bf A}$ denote its adjacency matrix and its Laplacian matrix, respectively. $\cal L_{\bf A}(i,i)$ is the degree of vertex ${{\bf{v}}_i}$, and $\mathcal L_{\bf A}(i,j) = -1$ if vertices…
Tony
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Why the null space of pseudo inverse equals the null space of the matrix transpose?

The pseudoinverse $A^+$ of A is the matrix for which $x = A^+Ax$ for all x in the row space of A. The nullspace of $A^+$ is the nullspace of $A^T$. I don't understand this cause the above seems to imply that $A^+=A^T$ which doesn't make sense as $x…
newbie125
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Pseudoinverse matrix and SVD

I'm trying to solve an homework question but I got stuck. Let A be a m x n matrix with the SVD $A = U \Sigma V^*$ and $A^+ = (A^* A)^{-1} A^*$ its pseudoinverse. I'm trying to get $A^+ = V \Sigma^{-1} U^*$, but I'm missing something. Can anyone help…
paulochf
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What is step by step logic of pinv (pseudoinverse)?

So we have a matrix $A$ size of $M \times N$ with elements $a_{i,j}$. What is a step by step algorithm that returns the Moore-Penrose inverse $A^+$ for a given $A$ (on level of manipulations/operations with $a_{i,j}$ elements, not vectors)?
Kabumbus
  • 438
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2 answers

L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value

Consider a matrix $A \in\mathbb R^{n\times m}$ with $n>m$. It has full column rank, i.e. $\operatorname{rank}(A)=m$. Its left pseudo-inverse is given by; $$A^{-1}_\text{left}=(A^TA)^{-1}A^T $$ From two different results during my studies, I have…
Metehan Yayla
  • 364
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What forms does the Moore-Penrose inverse take under systems with full rank, full column rank, and full row rank?

The normal form $ (A'A)x = A'b$ gives a solution to the least square problem. When $A$ has full rank $x = (A'A)^{-1}A'b$ is the least square solution. How can we show that the moore-penrose solves the least square problem and hence is equal to…
midi
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How can we compute Pseudoinverse for any Matrix

If we have a system of linear equations $Ax=b$, then in case $A$ is invertible it easy to say that the solution is $x=A^{-1}b$. In all other cases, even if $A$ is invertible, the solution if it exist will be in the form $x=A^{+}b+(I-A^{+}A)w$, where…
LoveMath
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Finding a minimal set of equations that determine a variable.

I have a system of $m$ linear equations on $n$ variables, which I'm representing as $Ax=b$, with $A$ an $m\times n$ matrix representing the equations and $b$ an $\mathbb R^m$ vector representing the constants of the equations. I'm given that there…
Rushabh Mehta
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Why SVD is not unique but the Moore-Penrose pseudo inverse is unique?

I feel confused about the uniqueness of the Moore-Penrose inverse generated from SVD. For any matrix $A$, if $X$ satisfied $$AXA=A, XAX=X, (AX)^\mathrm{T}=AX, (XA)^\mathrm{T}=XA $$then $X$ is called the Moore-Penrose inverse of $A$. If $A$ has the…
bregg
  • 125
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Tikhonov regularization vs truncated SVD

To find $\mathbf{x}$ such that $$A\mathbf{x}=\mathbf{b}$$ we can use least squares when the problem is not well posed. Further, we can use Tikhonov regularization when $A$ is ill-conditioned. In Tikhonov regularization, we minimize…
fragapanagos
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Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse of A. There is also a complex version, but I am…
Peter
  • 86,576
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Pseudo Inverse of product of Matrices

Let $A$ and $B$ are two matrices where $A \in \mathbb{R}^{m\times p}$ and $B \in \mathbb{R}^{p\times n}$ and both $A$ and $B$ are full rank matrices Now I really want to know in what cases $(AB)^+ = B^+A^+$ ,where $A^+$ is Moore-Penrose…
Cpt. Price
  • 121
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4 answers

What is the best way to compute the pseudoinverse of a matrix?

Mathematica gives the pseudo-inverse of a matrix almost instantaneously, so I suspect it is calculating the pseudo-inverse of a matrix not by doing singular value decomposition. Since the pseudo-inverse of a matrix is unique, is there a good formula…
nekodesu
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7
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Constructing matrices $A$ and $B$ such that $(A B)^+ \neq B^+ A^+$

How do I go about constructing two matrices $A$ and $B$ such that the pseudoinverse of $AB$ is not equal to the pseudoinverse of $B$ times the pseudoinverse of $A$?
user100503
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