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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What is the distribution of the elements of the Moore-Penrose inverse?

Assuming $A$ is an $m \times n$ matrix (with $n \ge m$) of normally distributed elements with $\mu_A = 0$ and $\sigma_A = 1$, is there a mathematical formulation for the distribution of the elements ...
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Does symmetry of $AB$ implies symmetry of $A^\dagger B$?

Let $A$, $B$ and $AB$ symmetric. Is $A^\dagger B$ also symmetric i.e. $$A^\dagger B = B A^\dagger$$, where $A^\dagger$ is the pseudo-inverse of $A$ ?
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On the existence of the Moore-Penrose inverse

The following was written in a paper, but I couldn't find out why. Does anyone have any idea on how to prove this claim? It is well known that $A^{\dagger}$ exists for a given $A \in B(H, K)$ if and ...
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Solution to equation with moore-penrose inverse

I have a linear equation of the form $$ C = (I-AA^+)X $$ where my variable is $X$ and $A$ is an hermitian operator and $A^+$ is the pseudo inverse. Assuming that the determinant of $A$ is $0$ (or ...
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Finding inverse of the operator

I am very new to finding inverses of the operators in the functional analysis. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a complicated ...
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Moore-Penrose pseudoinverse solves the least squares problem (SVD framework) [duplicate]

I am a computer science researcher who has to learn some numerical linear algebra for my work. I have been struggling with the SVD and Moore-Penrose pseudoinverse as of late. I am trying to solve some ...
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Relationship between cross-product and Moore-Penrose pseudoinverse [closed]

It is said in here https://blog.bham.ac.uk/intellimic/g-landini-software/colour-deconvolution-2/ that you can get the third vector of a 3x3 (stain) matrix either by taking the cross product of the ...
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How to calculate the generalized inverse of a matrix on new columns

Assume that there exists a matrix $A∈R^{m×n}(m≠n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula: $$ A=AXA\\ (XA)^T=XA $$ How to calculate the generalized inverse matrix Y of $...
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Generalized inverse of a matrix on new columns

Assume that there exists a matrix $A \in R^{m \times n}(m\neq n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula: $$ A=AXA\\ (AX)^T=AX $$ How to calculate the generalized ...
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Pseudo Inverse (Moore inverse) of the sum of two matrices (one of them has a rank of 1)

I have two matrices, A and B. A is a $m \times n$ matrix and B is a rank one matrix. is there a way to speed up the pseudo inverse (right inverse) of the sum of these two? I'll do this calculation ...
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Need help with solving a system of linear equations

Given $n$ samples of vectors $\vec{x}\in R^k$ and n corresponding ground-truths $\vec{y}\in R^k$ I need to find the least square solution $A\vec{x}+\vec{b}=\vec{y}$ i.e. solve for $A$ and $\vec{b}$ ...
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High-dimensional generalization of Penrose best approximate solution to least square equation.

Fix $A\in\mathbb{R}^{d\times d}$. The solution to $$ \min_{x\in\mathbb{R}^d: Ax=b}\Vert x \Vert_2^2 $$ is $A^{\dagger} b$. Roger Penrose showed (Corollary 1) that the the solution to $$ \min_{X\in\...
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Least squares projection with singular matrix

I am trying to better understand the solution of systems of linear equations (also in an $L_2$ sense for inconsistent ones) and for that purpose I have been trying to derive the set of solutions in ...
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how to prove $(A^2)^+=M^2$ if $M=A^+$ and A is a symmetric matrix?

Here $A^+$ is Moore–Penrose inverse. My question is: If $M=A^+$ and A is a symmetric matrix,then how to prove $(A^2)^+=M^2$?
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Example for singular positive semi definite matrix with positive diagonal and its generalized inverse has $0$ first element

I need to find a counter example that satisfies the following: 1- the matrix $A_{n*n}, n>2$ symmetric and positive semidefinite, and the main diagonal is positive $a_{ii}>0$ 2- the matrix $A$ is ...
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Eigenvalues of $A^+ A$

Let $A \in \mathrm{R}^{n\times m}$ be a matrix of rank $n$ (with $m>n$). Let $A^+$ be the Moore-Penrose pseudoinverse of $A$. The matrix $$ A^+ A = A^T(A A^T)^{-1} A $$ has eigenvalues $\lambda_i \...
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Simplify analytical expression that contains pseudoinverse and non-square matrices

I have a problem in a following form $$\mathrm{C = A\underbrace{X(YX)^{+}Y}_{\neq I}B}$$ where $+$ indicates pseudoinverse. $\mathrm{dim(C) = (L, 1) \\ dim(A) = (L, N) \\ dim(X)=(N, M) \\ dim(Y)=(M, N)...
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Proof of the pseudoinverse conditions

I am having a hard time understanding the pseudoinverse and I am not sure how to prove its most basic conditions. I know that $A\in\mathbb{R}^{mxn}$ with $m\geq n$ and $rank(A)=n$. I have to prove ...
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Expressing inverse of matrix product as product of pseudoinverses?

I have a full-rank $m \times n$ matrix $\mathbf{A}$ that is "broad" ($n<m$), and a symmetric $n\times n$ covariance $\mathbf{B}$ that is generally not full-rank. The symmetric matrix \...
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How to put restrictions on a closed form linear regression solution?

Consider task: $X_{n, 2}$ and $Y_{n,2}$ points, $Y= XR + \epsilon N(0, 1)$ where we want $R$ to be a rotation matrix. To find $R$ we may solve liner regression through pseudoinverse $T = (X^TX)^{-1}X^...
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Does $(I-BB^\dagger)C=0$ hold for a symmetric positive semidefinite matrix $G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$ with $C \preceq B$?

Given a symmetric positive semidefinite matrix $$ G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$$ where $A$, $B$ and $C$ are not invertible, and $C\preceq B$, does the following equality ...
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Find linearly dependent rows/columns of rank deficient matrix

I have the problem in the following form $C=AX(BX)^\dagger$ where $\dagger$ indicates pseudoinverse. $dim(A)=M, N$ $dim(B)=P, N$ $dim(X)=N, K$ with $M>N>K$ and $P>N>K$ All abovementioned ...
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How to find the pseudo inverse of a partitioned matrix by using the inverses of the matrix's blocks?

I am working with a partitioned matrix $A =\begin{bmatrix}E\\D\end{bmatrix}$. E and D have the same properties. $E\in R^{n\times n}$, $E$ is full column rank and nonsingular. $E = I\cdot f(\bar{x}) + \...
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Solving for matrix X (using the Moore-Penrose Pseudoinverse)

Consider the following matrix equation: $\mathbf{B'XB=A}$ where $\mathbf{X}$ is $m \times m$ $\mathbf{B}$ is $m \times n$ $\mathbf{A}$ is $n \times n$ $\mathbf{A}$ and $\mathbf{B}$ are known $\mathbf{...
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Pseudo inverse (left inverse) of $(I_n \otimes v^T) + (v^T \otimes I_n) $

Consider a column vector $v\in \mathbb{R}^n$. We are interested in finding the pseudo-inverse of the following matrix: \begin{align} A= (I_n \otimes v^T) + (v^T \otimes I_n) \end{align} where $I_n$ ...
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Confidence interval (error estimation) after Moore-Penrose pseudo inverse LSQ estimation

Context In engineering we can perform a parameter estimation on a system when the dynamics are linear w.r.t. the parameters: $$ f(\ddot{q}, \dot{q}, q) = \tau \\ \downarrow \\ \phi(\ddot{q}, \dot{q}, ...
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Prove that for every $\vec{y}\in R^n$ $AA^+\vec{y}$ is the orthogonal projection of $\vec{y}$ on the column space of $A$

Prove that for every $\vec{y}\in R^n$ $AA^+\vec{y}$ is the orthogonal projection of $\vec{y}$ on the column space of $A$ I have some intuition that this is true, but I have a hard time proving it. My ...
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Intuition about pseudoinverse

I am trying to gain intuition about the pseudoinverse through explaining what I know. Sometimes I feel like my "proof" of concepts are more about moving equations around instead of actual ...
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Expand Moore-Penrose pseudo inverse of two sum of symmetric matrices

I was studying a Moore-Penrose pseudo inverse matrix and found out below theorem in https://www.kybernetika.cz/content/1979/5/341/paper.pdf. If $V$ is $n \times n $ symmetrical matrix and if $X$ is an ...
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Proof, that B^T B is a regular matrix, when B is a set of basis vectors [duplicate]

im interested in following proof: Let $$B = [b_1, ... b_m] \in \mathbb{R}^{n\times m}$$ be a matrix, that consists of m independent basis vectors $$b_i$$ and m<n. Why is it, that $$B^TB$$ is a ...
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Find non unique solutions to pseudo inverse least square estimation

For the equation $Ax = B$, I can use the pseudo inverse of $A * B$ to get the best estimate for $x$. Now $A$ is not full rank and there's linearly dependent columns, so when performing $\operatorname{...
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Why is the pseudoinverse of an orthogonal projection matrix itself?

From both this paper and Wikipedia, it is mentioned that for an orthogonal projection matrix $(I - A^+A)$ its pseudo inverse is itself, i.e., $$(I - A^+A)^+ = I - A^+A$$ Why is this the case? Can ...
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Inverting the eigenvalues of the compact form of the eigendecomposition of a singular symmetric matrix

I have a symmetric matrix $\textbf{A}\in\mathbb{R}^{n\times n}$ of rank $r<n$. It can be eigendecomposed as $\textbf{A}=\textbf{U}\bf{\Lambda}\textbf{U}^T$ or equivalently as $\textbf{A}=\textbf{U}...
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Is $X^\dagger \Sigma (I-XX^\dagger)$ equal to zero?

Denote the Moore–Penrose inverse with $\dagger$. I would like to know if $X^\dagger \Sigma (I-XX^\dagger)$ equals zero, where $\Sigma$ is a full-rank diagonal matrix, and $X$ has full column rank for ...
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Find solutions to linear least square problem.

Find all solutions of the linear least squares problem $$\min_x\|Ax-b\|_2$$ where $A = \left[\begin{matrix} 1&1\\1&1\end{matrix}\right]$ and $b = \left[\begin{matrix} 1 \\ 2\end{matrix}\right]$...
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Verify that if $A\in\mathbb R^{m\times n}_n$, then $A^+ = R^{-1}Q_1^T$.

Given $A\in\mathbb R ^{m\times n}_n,Q^T\in\mathbb R^{ m\times m} , Q=\left[\begin{matrix} Q_1 & Q_2 \end{matrix} \right]$ orthogonal, $ R\in\mathbb R^{n\times n}_n$ upper triangular. Householder ...
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Proof of $\frac{1}{n}\mathrm{E} \left[ \| \mathbf{X}\mathbf{\hat{w}} - \mathbf{X}\mathbf{w}^{*} \|^{2}_{2} \right] = \sigma^{2}\frac{d}{n}$

I am trying to find a proof for the MSE of a linear regression: \begin{gather} \frac{1}{n}\mathrm{E} \left[ \| \mathbf{X}\mathbf{\hat{w}} - \mathbf{X}\mathbf{w}^{*} \|^{2}_{2} \right] = \sigma^{2}\...
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Matrix equation and pseudo-inverse

Let tall matrices $\mathbf{G}, \mathbf{U} \in \mathbb{R}^{m \times (m-k)}$ be full-column rank, where matrix $\mathbf{U}$ is semi-orthogonal, i.e., $\mathbf{U}^T \mathbf{U} = \mathbf{I}_{m-k}$. The ...
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Least-squares with an additional constraint of being real

To solve this: $$B_{opt}= \operatorname{argmin}\|Z-AB\|_2^2$$ We use the closed-form solution $$B_{opt}= (A^{H}A)^{-1}A^{H}Z$$ If $A$ and $Z$ have complex elements, $B_{opt}$ will be complex in ...
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Combining two linear matrix equations with non-square linear operators?

Assume I have two equations: $$\mathbf{x} = \mathbf{A}\mathbf{y}$$ where $\mathbf{x} \in \mathbb{R}^{m}$, $\mathbf{y} \in \mathbb{R}^{n}$, and $\mathbf{A}$ is a matrix of size $m$-by-$n$, as well as $$...
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If $P$ is an orthogonal projection, prove that $P^+ = P$.

If $P$ is an orthogonal projection, prove that $P^+ = P$. Where '$^+$' indicates the Moore-Penrose pseudo-inverse. What we know is the $P$ is symmetric and idempotent. That is $P = P^2 = P^T$. I am ...
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Negative dip near the diagonal of $K^{-1}$ where $K$ is a positive definite matrix

Say $K$ is a covariance matrix or a Gaussian process kernel matrix of $f(x_i)$. In other words, $K_{ij}$ is a covariance between $f(x_i)$ and $f(x_j)$ where $f(x)$ is a stochastic function. Assume ...
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Characterize all left inverses of a matrix $A\in\mathbb R^{m\times n}$.

Characterize all left inverses of a matrix $A\in\mathbb R^{m\times n}$. Note this for reference, we have characterized all right inverse of the matrix $A$. Observe that we solve for $AA_R=I_m$. Then $...
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Correlation between pseudo inverse and almost exact matrix

Let say I have 2 matrices (with a large dimensions), V, E. and $|V-E|_{2}\approx3\times10^{-11}$ I would expect that $VE^{+}=A$ would be very similar to the identity matrix up till replacement of some ...
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4 votes
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Characterize all solutions of the matrix equation $AX=B$ in terms of the SVD $A = U\Sigma V^T$.

Let $A\in\mathbb R^{m\times n}$, $B\in\mathbb R^{m\times k}$ and suppose $A$ has an SVD. Assuming $\mathcal R(B) \subseteq \mathcal R(A),$ characterize all solutions of the matrix linear equation $$AX ...
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2 answers
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For $A\in\mathbb{R}^{m\times n}$, prove that $\mathcal{R}(A^+) = \mathcal{R}(A^T)$.

For $A\in\mathbb{R}^{m\times n}$, prove that $\mathcal{R}(A^+) = \mathcal{R}(A^T)$. Attempt: Let $x\in\mathcal{R}(A^+)$, then $A^+y = x$ for $y\in\mathbb R^m$. But since $\mathcal{R}(A^T) = \mathcal N(...
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How to use Gaussian elimination to find the least squares pseudo inverse?

Rather specific but my professor showed me how to do this in class and now I forget how she did it... Her homework requires I do it this way. Also I cannot seem to find articles on the web explaining ...
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Generalized inverse of covariance matrix of conditional distribution of normal random vector

In this article of wiki, it says that if $(X_1',X_2')'$ is (multivariate) normal, the covariance matrix of $X_1|X_2=a$ is $\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}'$, where $\Sigma_{22}^{-1}$...
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show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$.

For $A\in\mathbb{R}^{p\times n}$ and $B\in\mathbb{R}^{m\times n}$, show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$. Note "$^+$" indicates Moore-Penrose pseudo-inverse. My ...
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Schur complement like operation on a singular matrix

For the classical definition of matrix inversion by Schur complement, given by: \begin{aligned} M^{-1}=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]^{-1} &=\left(\left[\begin{...
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