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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.

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Psuedoinverse of product of two symmetric matrices

I have two matrices, $A$ and $B$. $A$ is a symmetric blockwise diagonal matrix that can be represented as: $$ A=\begin{bmatrix} A_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \...
Ben's user avatar
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Bound for norm of projection onto $k$-dimensional subspace in terms of projections onto basis vectors

Let $V$ a finite-dimensional vector space with inner product $\langle, \rangle$. Let $v_1,\dots,v_k \in V$ linearly independent vectors of unit length. Let $p_i:V \rightarrow \text{span}(v_i)$ defined ...
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least squares minimum test error solution

assume we want to learn a model $y=x^T \beta + \varepsilon $ where $\beta \in \mathbb{R}^d$ is constant $ x \in \mathbb{R}^d$ is the input vector with Gaussian distribution $\mathcal{N}(0,\Sigma_x)$ ...
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An alternative to Levenberg–Marquardt algorithm

When trying to solve for a (over)determined non-linear least square method: $$\underset{x}{\min}||f(x)||^2_2, f: \mathbb{R}^n \rightarrow \mathbb{R}^m, x\in \mathbb{R}^n, m\geq n$$ we use the Gauss-...
William Lin's user avatar
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Solving for matrix in Ax=y

What can be said about the equation $Ax=y$ where $x,y$ are known vectors? I couldn't find much information about this equation online. The problem where $A,x$ are known is trivial by simply ...
Ervin Macić's user avatar
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Is the product of a right-invertible, an invertible and a left-invertible matrix itself invertible?

Suppose $B \in \mathbb{R}^{(n,n)}$ is invertible and $A \in \mathbb{R}^{(n,m)}$ is left-invertible. Is $A^T B A$ invertible? I know that $A^T A$ is invertible. I've been trying to work it out using ...
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Question about the induced two-norm of pseudo-inverse matrix

Given a full row rank matrix $A\in R^{m\times n}$, where $n>m$. Let $A^+$ be the pesudo-inverse matrix of $A$ and $a_{ij}$ be the element of this matrix, where $i\in\{1,\ldots,m\}$ and $j\in\{1,\...
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$(T^{*})^{+} = (T^{+})^{*}$

I'm trying to prove why $(T^{*})^{+} = (T^{+})^{*}$ only using properties of matrix operations (I'm considering $()^{*}$ and $()^{+}$ operations as well, just to be clear). However, I assume I cannot ...
pseudobulbose's user avatar
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Seeking a Linear Approximation for the Inverse of a Perturbed Pseudoinverse Matrix

I'm currently grappling with a challenge where I'm working with a matrix $L$, which is not of full rank, and its pseudoinverse $L^{+}$. My objective is to find a linear approximation for $$ \left(L^{+}...
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Moore-Penrose Pseudoinverse of Augmented Linear Systems of Equations

The problem I am working on is comprised of $N$ independent system of equations with the same size $$ A_{4096\times3}x_{3\times1}=b_{4096\times1} $$ where $x$ has to be found using Moore-Penrose ...
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Bounded (finite-rank) Inverse Operators

While studying for my functional analysis course I encountered bounded inverse theorem which states that given a bijective bounded linear operator $T:X\rightarrow Y$, then $T^{-1}:Y\rightarrow X$ is ...
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Does this matrix exist? A commonly encountered puzzle

Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of ...
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Show that if the rows of matrix can be split into two orthogonal linear subspaces, then the inverse has each pseudoinverse as columns

Assume an invertible matrix $C$ is in the form $\left[ \begin{matrix}A \\ B \end{matrix}\right]$, where the linear subspace generated by the rows of $A$ is orthogonal to the linear subspace generated ...
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Non-negative QP

I have a rather peculiar contrained quadratic programming problem where I try to fit a left-stochastic matrix and I am not sure how to properly solve it. Consider matrices $S\in\mathbb R^{D\times N}$, ...
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Pseudo-Inverse of mapping with identical null-spaces

I have two functions $f_a(x) = y_a$ and $f_t(x) = y_t$, where the dimension of the domain is larger than range. Finally, I want to compute the linearised mapping from $y_t$ to $y_a$ via ${}^t J_a = \...
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What is the property of matrix $S$ to be near-identity matrix for $M$, e.g. $SM$ is close to $M$?

I have a matrix $S$: ...
danbst's user avatar
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Question related to splitting of matrix

Let $A \in \mathbb{R}^{m \times n }; A=U-V$ be a splitting such that $R(A)=R(U), N(A)=N(U)$ where $R(A),N(A)$ denote the range space and null space $A.$ Definition of Moore-Penrose inverse of a matrix ...
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For what vectors does $A^+ A v = v$, where $A^+$ is the Moore-Penrose pseudoinverseof $A$?

Let $A$ be an $m \times n$ matrix of rank $r$, and let $A^+$ be its Moore-Penrose pseudoinverse. For what vectors $\vec{v}$ (of length $n$) is it true that $A^+ A \vec{v} = \vec{v}$? What about in the ...
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Solvability of $A X B=C$ with $X=X^\mathrm{T}$

I am studying symmetric solutions to the complex matrix equation \begin{equation} A X B=C, \end{equation} where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
Juan's user avatar
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Given a $n \times n$ matrix $A$, when do the first $p$ columns of $A^{-1}$ coincide with the Moore–Penrose inverse of the first $p$ rows of $A$?

I noticed that for ${A}= \left[\begin{matrix} \cos{\left(\phi\right)} & \cos{\left(\phi+2\pi/3\right)} & \cos{\left(\phi+4\pi/3\right)}\\ \sin{\left(\phi\right)} & \...
Fabio Dalla Libera's user avatar
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Orthogonal projection on a lower triangular non square block matrix $A \in \mathbb R^{m=4k,n=7k}$ such that $A_{i,j}\in \{0,1\}$

If i have a very large matrix matrix $$A \in \mathbb R^{m=4k,n=7k}$$ such that $$A_{i,j}\in \{0,1\}$$ and $A$ is a lower block matrix with diagonal block $$B_t\in \mathbb R^{4,7}$$ and block left of ...
D. Sikilai's user avatar
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Solving a matrix optimization problem (part 2)

Problem definition This post is a follow up of this previous one. Consider the following $m\times(n+1)$ matrix \begin{equation*} B(\sigma)\triangleq \left[\begin{array}{ccc} b_0^n(s_1) & \cdots &...
matteogost's user avatar
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Solving a matrix optimization problem

Consider the following optimization problem \begin{equation*} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert^2 \end{equation*} where $\sigma\in \mathbb{R}^m$ is the decision ...
matteogost's user avatar
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Continuity Properties of Pseudoinverse with Orthogonal Block

Let $p$ be a parameter taking values in a compact set $P$. Let $M(p) = \begin{bmatrix} A(p) & D \\\\ B & C\end{bmatrix}$ be a block matrix. Further impose that $A(p)$ is an orthogonal matrix ...
jarmill's user avatar
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Prove that the Moore-Penrose inverse of a symmetric matrix have the same nullspace as the original.

Prove that the Moore-Penrose inverse of a symmetric matrix have the same nullspace as the original. The definition in my textbook is as follows: Corresponding to any $m \times n$ matrix A, there is a ...
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Describe any vector $x$ that has uniform inner product with every vector in the given set.

Suppose we have a matrix $A\in \mathbb R^{n\times n}$ such that $\mathbf{1}\in \text{range}(A)$, where $\mathbf{1}$ is the all-ones vector. I want to find, explicitly with respect to the entries of $A$...
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Prove that $\text{tr}(B_1^{-1} B_2) \geq \text{tr}((A^\text{T} B_1 A)^{+} A^\text{T} B_2 A)$

Suppose that we have two real and positive definite $n \times n$ matrices $B_1$ and $B_2$ and that $A$ is an arbitrary real $n \times n$ matrix. Running some numerical tests by generating random ...
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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^...
Jord van Eldik's user avatar
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1 answer
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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 \...
Jie Wei's user avatar
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Pseudo-inverse matrix to minimise $L^1$ norm error

Given the over-determined 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 $...
Marco's user avatar
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4 votes
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144 views

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 ...
FreddyMro's user avatar
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1 answer
151 views

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 ...
Arian Ghasemi's user avatar
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2 answers
134 views

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 $A$. ...
clay's user avatar
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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, $B$ ...
Burak's user avatar
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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 ...
accounted4's user avatar
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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....
Guillem FN's user avatar
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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}}^{-...
PT_98's user avatar
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3 answers
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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 ...
Uri Cohen's user avatar
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6 votes
3 answers
316 views

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 ...
woody's user avatar
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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 ...
arcaynia's user avatar
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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 $J$...
Mike Battaglia's user avatar
1 vote
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93 views

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 — \\ \...
Jason Burton's user avatar
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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}, \...
Matthew James's user avatar
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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})^{...
Confounded's user avatar
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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 ...
HappyFace's user avatar
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3 votes
2 answers
158 views

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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1 answer
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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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