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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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Do I get better solution if I have more data - Pseudo Inverse

I wonder if I can get a better solution for this equation: $$Ax = b$$ If $A$ is not square and I use pseudo inverse $A^{\dagger}$ to find $x$ $$x = A^{\dagger}b$$ The reason why I asking this is ...
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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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1answer
51 views

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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19 views

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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28 views

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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1answer
30 views

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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1answer
42 views

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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1answer
55 views

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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1answer
40 views

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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1answer
36 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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1answer
42 views

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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1answer
79 views

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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1answer
32 views

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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35 views

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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30 views

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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1answer
70 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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1answer
40 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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1answer
63 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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1answer
81 views

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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30 views

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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1answer
41 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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29 views

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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1answer
58 views

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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39 views

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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50 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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1answer
29 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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1answer
110 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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1answer
189 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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1answer
35 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: $A B A = A$...
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1answer
39 views

Uniqueness of a pseudoinverse which is not a Moore-Penrose pseudo inverse

Definition Let $A \in M_{m,n}(\mathbb{R})$ be a matrix. Then a matrix $B \in M_{n,m}(\mathbb{R})$ is called a pseudoinverse of $A$ if we have $ABA = A$ and $BAB = B$. If in ...
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36 views

How to simplify matrix using pseudo inverse properties?

How to simplify the following equation $\mathbf{r^H D^{-1}p(p^HD^{-1}p)^{-1}p^H D^{-1}r }$ into Trace form or Norm form? Where $\mathbf{r}$ is $L\times1$ complex vector, $\mathbf{D}$ is $L \times L$ ...
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1answer
38 views

Which norm does pseudo-inverse for least square for matrix equation minimizes?

$$AX = B$$ where $A$, $X$, $B$ are general matrices with compatible dimensions. Does $$\hat{X} := A^+ B$$ minimize $\| AX - B \|_2$ or $\| AX-B \|_F$? Or both?
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1answer
32 views

if $A.x \le b$, when is $x \le A^{-1}.b$?

I have a relation, derived from optimality conditions of a linear program: $$ L \le A.x \le U$$ with: $L, U \in \mathbb{R}^{m}$ $A \in \mathbb{R}^{m \times n}$, with each element $\in \{0,1\}$ $x \in ...
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0answers
59 views

When does $\ker(M) = Span( I - M^{+}M)$ ? Always?

Let $M$ be an $m \times n$ matrix, where the rows are linearly independent. Let $M^{+}$ be the pseudo-inverse of $M$ (which I think is $M^{+} =M^T\left(MM^T\right)^{-1}$, at least when $m\leq n$). ...
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0answers
121 views

How to compute the minimum norm solutions of Matrix equation?

It is well-known that for the general linear system $$\mathbf{X}\mathbf{a}=\mathbf{y}$$ where $\mathbf{X}$ is a rectangular matrix and $\mathbf{a}$ and $\mathbf{y}$ are column vectors of compatible ...
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0answers
23 views

DLS inverse kinematics derivation via Lagrange Multipliers.

In my robotics book I have found the next justification for using Moore-Penrose inverse for inverse kinematics problem. Let us minimize the following functional $ g(\dot q) = \frac{1}{2} \dot{q}^T W \...
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43 views

Bott-Duffin-Inverse and linear equation systems

I am interested in checking feasiblity of linear equation systems of the form $$ \begin{split} A x &= b, \\ x &\geq 0 . \end{split} \tag{1} $$ I know that this is basically a linear program ...
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1answer
73 views

Derivative of an elementwise function and a pesudoinverse

I am trying to compute the partial derivatives of the following discrete time dynamic system with respect to $A$ and $B$ (for linearization purposes): $$ B_{t+1} = yf(A_tx)^\dagger\\ A_{t+1} = f^{-1}(...
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22 views

how to prove these theorems about MP-inverse?

Given the defining equations of MP inverse, (a) $AA^+A=A$ (b) $A^+AA^+=A^+$ (c) $(A^+A)^T=A^+A$ (d) $(AA^+)^T=AA^+$ where $A^+$ denotes the MP inverse of $A$, and several theorems: (i) $A^TAA^+...
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29 views

$(K\operatorname{*}K)^{\dagger}x = \sum_{n \in \mathbb N} {\sigma_n}^{-2} \langle x, v_n \rangle v_n$ where $({\sigma_n}^2,v_n,v_n)$ singular system

Let $X,Y$ be Hilbert spaces and $K$ be a compact operator and $(\sigma_n,u_n,v_n)$ a singular system of $K$. That is a decreasing sequence $\sigma_n \subset \mathbb R_+: \lim_{n \to \infty} \sigma_n = ...
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21 views

Block matrix inversion [duplicate]

In Wikipedia it is suggested that for any arbitrary sized A,B,C,and D one can use the block-wise inversion. It is also mentioned that A and D need to be square because they need to have an inverse. I ...
0
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1answer
21 views

What are the conditions to find $A$ such that $AB=BD$ where $D$ is block diagonal?

I have two known matrices $B\in\mathbb{R}^{m\times 9}$ and $D\in\mathbb{R}^{9\times9}$ where $D$ is block diagonal with each block written as $D_i=\alpha_iI_{3\times 3}$ and $I_{3\times 3}$ is the ...
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1answer
92 views

Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse.

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $. Below is ...