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.
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Moore–Penrose pseudoinverse and system of linear equations with infinitely many solutions
Is there an easy way to find out if a square system of linear equations has infinitely many solutions or no solutions at all by just using the Moore-Penrose pseudo inverse? I know how to do it by ...
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Validation of Normal Equations with Pseudo-Inverses
In an exercise for uni I am asked to prove that $w^* \in \mathbb{R}^d$ is a solution to $\min_{w \in \mathbb{R}^d}\lVert y - Xw \rVert_{2}^{2}$ if and only if $X^\dagger X w^* = X^\dagger y$, where $X^...
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One quick question about Moore–Penrose inverse of a symmetric matrix
I was wondering whether the following equation is correct, and why if so. Here the $
\left( A\right)^{+}
$ is the MP inverse of matrix A.
$$
\left(A^{\top} A\right)^{+}A^{\top} =\lim _{\rho \...
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Pseudo-inverse matrix to minimise $L^1$ norm error
Given the linear system $A\cdot x = b$, the least-squares solution (minimise $||A\cdot x - b||_2$) can be obtained by matrix multiplication:
$$x_\text{ls} = A^\dagger \cdot b $$
where $A^\dagger = (A^...
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Transform multiplicative noise to additive noise with singular matrices
I have a stochastic differential equation with multiplicative noise $\alpha(t)$
\begin{equation}
\dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}...
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Moore-Penrose inverse error approximation for matrix product
Given two matrices $A \in \mathbb{R^{m \times k}}, B \in \mathbb{R^{n \times k}}$ with $k> m > n$, I want to compute $(A B^{\dagger})^{\dagger} A$ where $\cdot^{\dagger}$ is the Moore-Penrose ...
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Procedure to generate an inverse for a non-square matrix
We know that if a matrix is $m \times n$ where $m=n$ then it has an inverse if its full rank.
But the interesting fact for me is about non-square matrices, where $m\neq n$, and also I searched on the ...
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For ($n \times p$) $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} $, show that $A_{22} = A_{21}A_{11}^{-1}A_{12}$
I'm currently trying to solve the following problem.
$A$ is $n \times p$ matrix with rank $r < \text{min}(n,p)$ and $A$ is partitioned as follows. $$A =
\begin{bmatrix}
A_{11} & A_{12} \\
A_{...
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For $m \times n$ matrix $A$ with $m \ge n$, show ${\lVert A^+ \rVert}_2 = 1/\sigma_n$ where $A^+ = (A^*A)^{-1} A^*$ [duplicate]
For $m \times n$ matrix $A$ with $m \ge n$, show that the norm of the pseudoinverse ${\lVert A^+ \rVert}_2 = 1/\sigma_n$ where $A^+ = (A^*A)^{-1} A^*$ and $\sigma_n$ is the nth singular value of $A$.
...
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Simplifying a matrix expression that involves a Moore-Penrose pseudoinverse
For $m\geq n$, suppose $L$ is an $m\times n$ real matrix and $P$ is an $m\times m$ diagonal matrix with nonnegative entries. I did some numerical experimentation to verify that $L'P (PLL'PLL'P)^\...
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Use of the inequality $A \succeq B^{\dagger}$ when B is not invertible
In semi-definite programming (SDP), you might have an optimization problem where $A \succeq B^{-1}⪰0$ is a constraint, which implies that $A_{ii} \geq (B^{-1})_{ii}$ for all $i$. In some cases, $B$ ...
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Solving for a dot product operand
The vector $y$ contains the dot products of vector $x$ against each row of matrix $A$. It can be expressed as follows:
$$y = Ax$$
$y$ and $A$ are given in my problem, and I am solving for $x$. My ...
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Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns
Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i....
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L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value formal source?
Given a full column rank matrix $A \in \mathbb{R}^{n\times m}$. The left pseudo-inverse is $A_{\text{left}}^{-1}=(A^\top A)^{-1}A^{\top}$. Then we have the following relationship
$$\|A_{\text{left}}^{-...
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Solve a linear equation XA=B using matrix with zero diagonal
Consider the following problem: given $A,B\in\mathbb{R}^{N\times P}$ for $N>P$, we wish to find $X\in\mathbb{R}^{N\times N}$ such that:
$$XA=B$$
Without further constraint, the solution is given by ...
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How to prove $\mathbf{1}^\top\mathbf{Q}^+\mathbf{Q}=\mathbf{1}^\top$, where $\mathbf{Q}$ is any element-wise squared correlation matrix?
Let $(X_1,…,X_n)$ be a random vector with $0<\prod_{j=1}^n\text{Var}(X_j)<∞$.
Let $\mathbf{Q}=(\mathbf{q}_{1},…,\mathbf{q}_{n})=(ρ_{jk}^2)_{n×n}$, where $ρ_{jk}$ is the Pearson correlation ...
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Potential error in paper asserting commutativity of symmetric matrix with Moore-Penrose projection operator
EDIT: The question now includes typeset equations rather than screenshots of the paper and further details. I have also rephrased the question as since I originally asked it I have done more ...
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Vector similar to, but not quite equal to least squares approximation of another vector
Suppose we have some vector $V$, and we want to locate the vector $kV$ which best approximates some other vector $J$. The answer $T$, in the least-squares sense, is the vector you get by projecting $J$...
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Are least square solutions maintained?
If I have a least squares problem $AXB = C$ for $X$, where $BB^+ = I$ ($B^+$ being the Moore-Penrose pseudo-inverse). Does this last property make it so that the first least sqaures problem and the ...
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SVD-based pseudoinverse solution sensitivity to equation linear combinations
My problem is of theoretical nature. Given an overdetermined system of $m$ equations in $n$ unknowns, $\bf A x = b$, where $m \gg n$ and
$$ {\bf A} = \begin{bmatrix} — {\bf a}_1 — \\ — {\bf a}_1 — \\ \...
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Solve nonlinear algebra equation
I'm trying to find the best way to solve this equation for the vector $x$.
\begin{equation}
Ax+b-ty=0
\end{equation}
Where $x=[x_1, x_2, x_3, ... , x_n]^T$ and $y = [\frac{1}{x_1}, \frac{1}{x_2}, \...
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Linear least squares and homogeneous solutions
When I studied regression using linear least squares (LLS), the solution to $\mathbf{X}\boldsymbol{\beta} = \mathbf{y}$ was always presented as
$$\boldsymbol{\hat{\beta}} = (\mathbf{X}^T\mathbf{X})^{...
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What can be said about $AA^+$ for symmetric real matrices?
Defining $A^+$ as the pseudoinverse of matrix $A$, what can be said about $AA^+$ if $A$ is a real matrix? What if $A$ is also symmetric?
The reason I am asking this is that I want to see why the ...
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Moore-Penrose pseudoinverse as a limit
For any matrix $A \in \mathbb{C}^{m \times n}$, there exists a unique matrix $A^{+}$ such that:
$$A^{+} A = \left( A^{+} A \right)^{*}, \qquad A A^{+} = \left( A A^{+} \right)^{*}, \qquad AA^{+}A=A, \...
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Question regarding the projection of a vector onto the nullspace of a matrix
So for a matrix $A$, we have that it's nullspace is spanned by the columns of $(1-A^+A)$ where $A^+$ is the pseudo inverse of $A$. I am pretty sure the projection of a vector $v$ onto the nullspace of ...
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Integral involving a Dirac delta
The following result is provided in the Matrix Cookbook (Equation 547): If $\mathbf{A}$ is an $m \times k$ matrix with linearly independent columns and $m > k$, then
\begin{align}
\int f(\mathbf{x})...
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Finding the nearest pseudoinverse pair for two matrices
Given matrices $A$ and $B$, I want to find "nearby" matrices $A'$ and $B'$ that constitute a pseudoinverse pair. More precisely, given $A \in \mathbb{R}^{n\times m}$ and $B\in \mathbb{R}^{m\...
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Polar Decomposition using Pseudoinverse
It is well known that every operator $A$ on a finite-dimensional space has a polar decomposition $A = UP$ where $U$ is a unitary operator and $P = \sqrt{A^*A}$. Further, when $A$ is not invertible the ...
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Is it true that $\lim\limits_{z \to 0^+} \left((A + z I)^{-1} + B\right)^{-1} = A(A + ABA)^+ A$?
Is it true that $\lim\limits_{z\to 0^+} ((A + zI)^{-1} + B)^{-1} = A(A + ABA)^+ A$,
where $A, B$ are real symmetric positive semidefinite matrices, and $P^+$ denotes Moore-Penrose pseudo inverse?
...
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Is $A - AB(B + BAB)^+ BA = A(A + ABA)^+A$ true for positive semidefinite $A, B$?
From extensive numerical simulation it seems that the following identity holds for two symmetric positive semidefinite matrices:
$$
A - AB(B + BAB)^+ BA = A(A + ABA)^+A.
$$
I tried to prove this, and ...
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SVD: Picking U and V when singular values are repeated
Is there a correct/stable way of dealing with repeated eigenvalues in S?
I was working on a Moore-Penrose Inverse in a library for a programming language at work. The dummy matrix I picked turned out ...
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Derivation of the matrix condition number of linear equations
When I was reading the derivation of the matrix condition number for linear equations on Wikipedia, $||\textbf{b}||$ can be directly replaced by $||\textbf{Ax}||$ as the non-singular square matrix is ...
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Why is $(A^+ A)^\ast = A^+ A$
Let $A$ be an $m\times n$ matrix with coefficients in $\mathbb R$. Supposedly this fact is true, however when I compute it:
\begin{align}
(A^+ A)^\ast &= A^\ast (A^+)^\ast \newline&= A^\ast ((...
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Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger $
Let $A$ be an $m\times n$ matrix with coefficients in $\mathbb{C}$. Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger $ where $A^\dagger$ is the Moore-Penrose Pseudoinverse of $A$.
My attempt:
\begin{...
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Projection with a system involving a singular matrix
Assume that I have a system given as
$$
\begin{bmatrix}
A& -B\\
0 & C
\end{bmatrix}
\begin{bmatrix}
A^{-1}& 0\\
0 & I\\
\end{bmatrix}
\begin{bmatrix}
D & B\\
0 & I\\
\end{...
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Prove: if $\bf{AB^T}$ is skew-symmetric and $\bf A$ full-rank, then $\bf{AX}=\bf B$ has unique solution $\bf X$
I've run into this statement while trying to prove that the energy of a rotating body in $N$ dimensions is conserved, it's the last puzzle piece I'm missing.
Let $\bf A$ and $\bf B$ be two $M \times N$...
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singular value decomposition of sum
Let $A,B$ be positive, linear trace class operators on some Hilbert space.
I would like to know if the following trace inequality for some $\mu>0$ is true
$$
\mathrm{Tr}\!\left(A\,(A+B+\mu I)^{-1}\...
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About a Matrix equation $ P(LP)^\dagger = (LP)^\dagger$
Have solved!
According to @obareey's comment, the key to the proof is trying to verify $P(LP)^\dagger$ is just the pseudoinverse of $LP$, using the existence and uniqueness of Moore-Penrose ...
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choosing the correct matrix to solve this optimization problem - Pseudoinverse
While searching for the uses of the pseudoinverse i stambled across this problem:
The way i approached it
$u_i = f_i \Delta t + u_{i-1}= x_i\Delta t + u_{i-1}$ and $s_i = 0.5 x_i \Delta t^2 + s_{i-1}$...
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Is there a way to visualise and pick solutions derived from multi-solution matrix systems?
I've been working with Moore-Penrose pseudoinverse matrices in the same practical/applied engineering system which I've referred to in a previous question to get solutions where matrix inverses cannot ...
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Calculate $\mathbf{A}^+\mathbf{y}$ indirectly without accessing $\mathbf{A}$
The Original Question (informal):
I have a black-box linear system $f:\mathbb{R}^N\rightarrow\mathbb{R}^M~(0\ll M\ll N)$. It is guaranteed that there exists a full-rank matrix $\mathbf{A}\in\mathbb{R}^...
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Left pseudo inverse of jacobian matrix
I am trying to implement the Gauss-Newton algorithm according to the wikipedia article: https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm
In here, the author speaks about calculating the ...
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Pseudoinverse of Gaussian matrix converges to its transpose
Let $X$ be an $n\times m$ matrix with iid $\mathcal{N}(0,1)$ entries.
I would like to show that
$$X^T\left(XX^T\right)^{-1}\rightarrow X^T$$
as $m\rightarrow \infty$, and $n$ is held fixed (and small)....
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Symmetric vector least squares solution
I have the similar problem as the Symmetric linear least squares solution. The least square problem of mine is that I want to find
$$
minimize || Ax-b ||^2,
$$
$$
where A\in m\times n,
$$
$$
x,b \...
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Find the closest point to $b= (1,1,2,-2)^T$ spanned by three vectors
EDIT: @Will Jagy suggests that $A$ doesn't consist of linearly independent vectors. Should I use the column space of the $A$ instead?
Find the closest point and the distance from
$$b=
\begin{pmatrix}
...
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Least Squares with Linear Algebra, Can't find the inverse of A^T * A, is there any workaround?
I'm currently creating a program for a class that uses linear algebra to perform linear and non-linear regression. using $\hat{x} = (A^T*A)^{-1}*A^T*Y$. If i have n points, I should be able to find a ...
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Determinant of product of matrix and nullspace
Assume I have a symmetric, positive-definite matrix $S \in \mathbb{R}^{p \times p}$. Assume that there is some matrix $L \in \mathbb{R}^{n \times p}$ that has full row-rank, i.e., has rank $n$ and ...
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Regular semigroups -- intuition
I'm trying to develop some intuition around the definition of the pseudoinverse in a regular semigroup.
Let the semigroup be $S$ with its associative operation written by juxtaposition. The ...
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Show that the Moore-Penrose pseudoinverse is the only unique left inverse of a non-square matrix
Let $A$ be an $N \times d$ matrix with $N > d$. Let $A^+$ be the Moore-Penrose pseudoinverse, i.e., $$A^+ = \left( A^T A \right)^{-1} A^T$$ We can see that $A^+$ is a left inverse of $A$ as $A^+A=I$...
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using null-space for positivity
I have the rectangular matrix $M$ and its pseudoinverse $M^{-1}$ as well as a vector $v$.
Using them, I find the vector $w = M v$. Since $M$ is rectangular and thus noninvertible, there has to be ...
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