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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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Existence of left or right pseudoinverse

I'm a bit confused about the following question: Given a matrix $A \in \mathbb{R}^{2 \times 3}$, which one of the left or right pseudoinverse does exist? I know that the left pseudoinverse exists, ...
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Distribution of a Complex Normal Pseudo-inverse matrix

I'm working on a problem where I have a random matrix $\mathbf{W} \in \mathbb{C}^{N \times M}$ and its elements are i.i.d. $w_{nm} \sim \mathcal{CN}(0, \sigma^2)$. I'm interested in the statistical ...
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Pseudo Inverse of a Seperable Transform

I have a problem which is of the form $ Y = AXB $, Where $Y$,$A,X,B$ are matrices of the form $Y \in C^{m \times n} $ , $A \in C^{m \times p}, X \in C^{p\times p},X \in C^{p\times n} $ . Can the ...
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About the pseudoinverse $A^{+}$ in Gilbert Strang's “Linear Algebra and its Applications 2nd Edition”.

I am reading Gilbert Strang's "Linear Algebra and its Applications 2nd Edition". He wrote "All solutions of $A \overline{x} = p$ share this same component $\overline{x_r}$ in the row space, and ...
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Moore-Penrose pseudoinverse of hermitian matrix

So I know that the least squares solution of smallest norm of the linear system $Ax = b$ where A is an $m \times n$ matrix is given with the pseudoinverse $x^* = A^{+}b = UD^{+}V^{T} b$ where the ...
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Pseudo inverse of a singular matrix without any linearly independent rows or columns

It is given here that "when A has linearly independent columns (and thus matrix $A^{*}A$ is invertible), $A^{+}$ can be computed as: $A^{+}=(A^{*}A)^{-1}A^{*}$ " and there is a similar expression for ...
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Question about convexity of least-squares problem and pseudoinverse

In the rank-deficient case, for A $\in \mathbb{C}^{m\times n}$, the solution of the least-squares problem, with b $\in \mathbb{C}^m $, $$min_{x \in \mathbb{C}}||Ax-b||_2$$ is not unique. (i) Prove ...
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difference of two orthogonal projections is orthogonal projection

Premise: I have an $n × q$ matrix $X$ and a $q × a$ matrix $C$ with $n > q > a$. I'm interested in the structure of the matrix $$ M = X X^+ - X_0 X_0^+ $$ where the superscript $^+$ indicates ...
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Matrix inversion fails, using either inverse or pseudo inverse!

I am doing this little manipulation on Matlab and I really don't understand why it doesn't work. Here it is: First I generate a $1\times400$ vector A, and a $400\times3$ matrix C. Then I generate a $...
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Factorization of Square-integrable random-variables and Generalized Inverses

Suppose that $X,Y,Z \in L^2(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^d)$, are $d$-dimensional random-vectors and there exists functions $f,g\in L^2(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),Law(X);\...
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Pseudoinverse of pseudoinverse of matrix A equals A: ${(A^{+})}^{+}=A$

As you know, a Matrix $A^{+}\in \mathbb{R}^{m\times n}$ is called a a pseudoinverse of $A\in \mathbb{R}^{n\times m}$ if $\Vert b-A A^{+} b \Vert_2\leq\Vert b- A y\Vert_2 \forall b\in \mathbb{R}^{n} \...
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Pseudoinverse of a matrix?

In pseudoinverse of a matrix, we have a special case when the columns are linearly independent. It is mentioned in that article and in other articles that It follows that $A^+$ is then a left ...
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Verifying the last Two Moore-Penrose Equations

If A is an m x n matrix with rank(A) = n, then $A^{+} = (A^{T}A)^{-1}A^{T}$. I already proved the first two of the Moore-Penrose equations. The second two are to verify: 3) $(AA^{+})^{*} = AA^{+}...
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What's the reason to use Singular Value Decomposition instead io $(A^TA)^{-1}A^T$ for pseudo inverse?

I wonder what's the reason to use this formula from Singular Value Decomposition $$ A = U\Sigma V $$ $$ A^{\dagger} = V\Sigma^{-1}U^T $$ Instead of $$ A^{\dagger} = (A^TA)^{-1}A^T $$ Both give ...
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Left PseudoInverse and Right Pseudoinverse in Linear Algebra

While reading pseudoinverse topic in Linear algebra i got confused in following Statements : 1) Left Pseudoinverse of a matrix is Projection onto Row space of a matrix. 2) Right Pseudoinverse of a ...
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Projection of a vector into the nullspace of a matrix

I need a clarification about the correct way to compute the projection of a vector into the nullspace of a matrix. For sake of clarity, let's call $A$ the matrix, $N(A)$ it's kernel and $A^\sharp$ ...
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Should I use pseudo-inverses to prove this?

Let $A=A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ and $z≥0$. Let $A(x)y≫0$ for $x≠x^*$. Then $z≯0$. How to prove this? Should I use pseudo-inverses?
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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 ...
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The definition of orthogonal complement in the column space

Denote $A^\bot$ is the matrix satisfied $A'A^\bot=0$ with the highest rank. Proof that: (1)$I-(A')^-A'$ is a $A^\bot$, here $A^-$ means pseudo inverse. (2)$M(A^\bot)=M(A)^\bot$. $M(A)$ is the column ...
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Relation between pseudo inverse and eigenbasis of a matrix

The question is if the Pseudo-Inverse of a matrix A is the same as the transformationmatrix in to the Eigenbasis of A. I fail to see any connection between the pseudoinverse of A matrix and its ...
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How can I estimate an first ODE with first order data, with the third order ODE form?

Let's assume that we have the data $y(t), u(t)$ and it's from a first order ODE: $$ \dot y(t) + a y(t) = b u(t) $$ But we have a ODE form at third order: $$ \dddot y(t) + a \ddot y(t) + b \dot y(t) +...
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Frobenius norm of $||AA^+ - I||_F = ? $

I need to find a value for the following norm $||AA^+ - I||_F$, where: $A^+$ is the Moore–Penrose Inverse matrix $||A||_F = \sqrt{Tr(AA^T)}$ A have $n \times m$ dimension and have rank $r$ I have ...
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matlab - Optimization with inverse and pseudoinverse

Let us assume I have to optimize this system: $$\min_{x\in S} \left|\left|\left(E\begin{bmatrix} I_n\\ A'(x)^{-1}C'(x)\\ \end{bmatrix}\right)^+ a -b\right|\right|^{2}$$ Where x is the vector ...
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why $x = \mathbf A^{\dagger}b$ is the one that minimizes $|x|$ among all mimizers of $|\mathbf Ax - b|$

for arbitrary matrix $\mathbf A\in \mathbb R^{m \times n}$ and $rank(\mathbf A) = r$, solve the least squares: $$\min \|\mathbf Ax - b\|_2. $$ According to SVD, pseudo inverse of $\mathbf A$ is $$\...
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Projection matrices $\mathbf{A}^{+}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{+}$

We are learning about pseudoinverses using the Strang book and I am just confused as to how to interpret the pseudoinverse. How come $\mathbf{A}^{+}\mathbf{A}$ projects into row space and $\mathbf{A}...
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39 views

Sum of matrix series

I have a matrix $M = \begin{pmatrix}\frac{1}{2}&\frac{2}{9}&0\\\frac{4}{9}&\frac{5}{9}&0\\\frac{1}{18}&\frac{2}{9}&1\end{pmatrix}$. I want to compute the sum $$\mathrm{E} = \...
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Fast Pseudoinverse of Block Lower Triangular Matrix

Let M be a wide matrix (more columns than rows) $$ M =\begin{bmatrix} C & 0 & 0 & \dots & 0\\ CA & C & 0 & \dots & 0\\ CA^2 & CA & C & \dots & \vdots\\ ...
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Derivative of the pseudoinverse

Let $K$ be an $n \times m$ matrix with $rank(K)=m$ and consider the pseudoinverse $K^+=(K^TK)^{-1}K^T$. What is the derivative of $K^+$ with respect to some scalar parameter $p$ (Derivative of the ...
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An example of the Pseudo-inverse of an operator

Let $E$ an infinite dimensional complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. Definition: Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of ...
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Cheap Implementation for the Moore Penrose Pseudoinverse

Can some one explain Adam W's implementation for finding the Moore Penrose Pseudoinverse? http://math.stackexchange.com/questions/75789/what-is-step-by-step-logic-of-pinv-pseudoinverse/317053#317053 ...
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Solving $MM^\dagger$ when with a Row Partitioned Pseudoinverse

Let $$ M = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^k\end{bmatrix}\\ Y = \begin{bmatrix} y_0 \\ \vdots \\ y_k\end{bmatrix} $$ Is there a convenient form to compute $MM^\dagger Y$ with $M^\...
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Book recommendation for matrix pseudoinverses?

Can any of you recommend a solid book on generalised matrix inverses, specifically the Moore-Penrose inverse? I have a good background in linear algebra, but little in numerical methods and i was ...
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Relation between Laplace solution of differential equations and the Pseudo-inverse

I am trying to understand the relation between the solution of differential equation in Laplace space and matrix inverse/pseudo-inverse problems. Consider the system of ODEs: $$\dot{\mathbf{x}}(t) = ...
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Model-based Pose Estimation using Jacobian and Pseudo Inverse

following this lecture, I would like to ask for clarification in two specific steps I am not being able to solve by myself. I will write in detail about my question after introducing where a possible ...
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uniqueness of pseudoinverse

I am reading Linear Algebra by Friedberg. In page 414 chapter 6.7, the first paragraph of the section of pseudoinverse of a matrix writes Let $A$ be an $m\times n$ matrix. Then there exists a unique $...
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Solution of linear system of equations with minimum norm.

In least squares estimation when $X^tX$ is not full rank, a solution can be found using the Moore Penrose pseudo Inverse. The Moore Penrose pseudo inverse of $X^tX$ gives a solution of the linear ...
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127 views

Moore-Penrose psuedoinverse of Laplacian

I am trying to attain the Moore-Penrose psuedoinverse of a a very large, sparse, rank degenerate, singular, and square matrix. (75000x75000, near rank) I realize the inverse will be very dense. I have ...
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67 views

Volume and barycentric coordinates of $k$-simplex in $\Bbb{R}^{n}$

How can the volume and barycentric coordinates (aka area/triangular coordinates) of a $k$-simplex in $\Bbb{R}^{n}$ be calculated given the vertices? In general $k \le n$ but any special cases for $k=n$...
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102 views

Pseudo-inverse Alternative Forms

In the book Convex Optimization by Stephen Boyd on page 649, the pseudo-inverse is defined as: $A^{\dagger}=V\Sigma^{-1}U^T$ this is the SVD decomposition. After that it says alternative forms are: ...
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Solving inequality-constrained least-norm problem non-iteratively

I am looking into the following constrained quadratic program in $x \in \mathcal{R}^4$ $$\begin{array}{ll} \text{minimize} & \| x \|_2^2\\ \text{subject to} & A x = b\\ & x_{\min} \le x_i ...
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Recursive generalized pseudoinverse

Let $A\in\mathbb{R}^{\text{n$\times$m}}$ be a rectangular matrix, and $n,m\in\mathbb{Z}^{+}$. The generalized pseudo-inverse $A^g$ is defined as $AA^gA = A$. The computation complexity of the ...
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56 views

Find the slope of the line using least squares method

This question is taken from GATE 2005 ICE subject paper. Using the given data points given below, a straight line passing through the origin is fitted using least squares method. $$(x, y)$$ ...
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Pseudo-inverse from CUR decomposition

Given a matrix A, I know there is a way to get its pseudo-inverse (a.k.a. Moore-Penrose inverse) from its SVD. If $A = USV^T$, then $A^{\dagger} = VS^{\dagger}U^T$ Let's, instead, imagine I know a ...
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Pseudo inverse solution of $(A\circ B)x=v$ if A is singular and B is symmetric

Given A a singular matrix, and B a symmetric matrix, I find that solving $(A\circ B)x=v$ by $$ x= (A\circ B)^+v $$ gives the correct solution to my physical system ($\circ$ denotes Hadamard product). ...
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Pseudoinverse solution of $(A+B)x=v$ when $A$ is singluar and symmetric and $B$ is symmetric

Given $A$ a singular and symmetric matrix, and $B$ a symmetric matrix, I find that solving $(A+B)x=v$ by $$ x= (A+B)^+v $$ gives the correct solution to my physical system. The physical system 1) ...
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66 views

Finding the pseudo inverse of a function

Consider the following matrix function defined: $$D_X(Y) = XYX^\top - \omega Y\omega^\top.$$ I have $D_\Gamma(L) = \frac{\partial}{\partial\vartheta}\Gamma$. Using the pseudo inverse of $D_\Gamma$, ...
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32 views

Condition for $\|X_n^{-1}\|$ to be bounded?

Let $\lambda_n = \|X_n^{-1}\|$, where $X_n$ is a non-singular $p\times p$ square matrix and $\|A\| = \sup_{|x| = 1}|Ax|$, with $|\cdot|$ the Euclidian norm. Is there a sufficient condition so that the ...
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209 views

Proof for Moore-Penrose inverse of transposed matrix

I would like to ask how to prove that Moore-Penrose inverse of $A^T$ is $(A^+)^T$. I know that I can do it by proving all 4 properties but I am stuck at proving that $(A^+A)^T=A^+A$ and $(AA^+)^T=AA^+...
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263 views

Relation between the maximum eigenvalues of symmetric positive definite matrix $A$ and $B A B^\dagger$

Suppose $A$ is a symmetric positive definite matrix of $n \times n$ dimensions. Matrix $B \in \mathbb{R}^{m \times n}$ is a full-ranked real-valued matrix with $m$ strictly smaller than $n$, i.e., $m ...
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40 views

Do fewer axioms suffice to define the Moore-Penrose pseudoinverse? (motivated by least squares method and group theory)

Definition of the Moore-Penrose pseudoinverse We know that each matrix $A \in M_{m,n}(\mathbb{R})$ has a unique matrix $B \in M_{n,m}(\mathbb{R})$ that suffices the following four axioms: $A B A = A$...