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Questions tagged [svd]

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.

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What is the accuracy of SVD in 3d transformations

I have a triangle $x$ with points $x_1,x_2,x_3\in\mathbb{R}^3$ that were measured at one location. The triangle was then transformed to $\bar x$ where the points were measured again as $\bar{x}_1,\bar{...
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Additivity of nuclear norm for projections

Let $A,B\in \mathbb R^{m\times K}$ with $B=U\Sigma V^T$. Let $r=\operatorname{rank} B$, $(u_1,\ldots,u_m)$ be the columns of $U$, and $S_1=\operatorname{span}(u_1,\ldots,u_r)$. Similarly let $(v_1,\...
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Using SVD to generate a transformation for calibration

Given the problem of trying to find a transformation matrix from one camera to another on a vehicle in a calibration room with the usual checkerboard floor and walls, we can capture images from both ...
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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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On an inequality involving operator norm of matrices and singular value

Let $A, E \in M_n(\mathbb C)$ be as in this question On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$ . How to prove that $\dfrac {||A^{-1}b-(A+E)...
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On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$

Let $A,E \in M_n(\mathbb C)$ . Suppose $\sigma_\min >0$ be the smallest singular value of $A$ and $||E||_2 < \sigma_\min$. Suppose $||A^{-1}E||_2 <1$. Then how to show that $A+E$ is ...
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$A \in \mathbb{C}^{m\times n}$,$A=FG^*$ and $r(A)=r(F)=r(G)$. Prove $A^\dagger = G(F^*AG)^{-1}F^*$ and $A^\dagger = (G^\dagger)^*F^\dagger$

Let $A^\dagger$ be a Moore-Penrose inverse of a matrix $A$. If $A \in \mathbb{C}^{m\times n}$ and $A=FG^*$, for some $F,G$ and $r(A)=r(F)=r(G)$, prove that $$A^\dagger = G(F^*AG)^{-1}F^*$$ and $$A^\...
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Full and reduced SVD of a 3x3 matrix.

I currently studying for an exam, and I'm currently working my way through some old exam problems and I'm currently at the following. First, we have a matrix $A= \begin{bmatrix} 2&0&0\\2&...
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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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Search for projection on a special matrix space with regard to Frobenius norm(computer vision background)

Background Define essential space as $$\varepsilon=\{E \in \mathbb R^{3\times3}|E=\hat{T}R\}$$ $$\hat{T}\in\{S\in \mathbb R^{3\times3}|S^T=-S\}$$ $$R\in\{A\in\mathbb R^{3\times3}|A^TA=I,\det(A)=1\}$$...
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Decomposition of a symmetric matrix $xx^T$ into a rank one and residual matrix?

Suppose we have $M=xx^T$ where $x$ is a random vector in $\mathbb{R}^n$. Also, we know that $x=q+e$ where $q$ is distributed according to $D$, i.e., $q \sim D$ and $e$ is a bounded vector. Therefore, $...
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Singular Values of Symmetric Matrix

I saw the following claim in this thread: How to compute the SVD of a symmetric matrix? Claim: The singular values of a symmetric matrix $A$ are the absolute values of its eigenvalues. I ...
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Relation between SVD and POD

I understand that POD is about choosing an optimal base, and i have found this Eckart Young theorem And also have encountered on a book that given an matrix $A$ its projection on POD modes is given ...
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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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How to deal with the non-uniqueness of SVD in numerical applications?

There are many applications in applied mathematics where the SVD of a matrix comes in handy. For example, consider the problem where we want to find an approximate solution to a(n) (overdetermined) ...
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SVD of a specific matrix and Singular values behaviour

I have a rectangular matrix $A \in \mathcal{M}_{l,n} (\mathbb{C})$, $l>n$ which has this property : $$A=\left[\begin{matrix} M_1 & i M_2 \\ M_2 & -i M_1 \end{matrix}\right]$$ Where $M_i$ ...
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constrained rank approximation

I'm trying to solve a problem similar to this problem. Instead of requiring the diagonals to be 0, I'd like to require columns of the low rank approximation to decrease in value while going down the ...
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Properties of singular value decomposition

Every (real) $m\times n$ matrix $A$ of rank $r$ has an SVD $$ A = U\Sigma V^T $$ Now, I have read about the following properties: $\text{Image}(A) = \text{span}\{u_1,\dots,u_r\}$ $\text{Null space}(...
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Why SVD matrix is orthogonal matrix?

I am trying to understand Why SVD matrix is orthogonal matrix. Let define SVD as A=UDV', I want to prove that U is a orthogonal matrix. I understand each column of U is the eigen vector of AA'. I ...
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Range space $\mathcal{R}(\textbf{A})$ the same as $\mathcal{R}(\textbf{AA}^H)$?

I'm working on a problem as follows: Given $\textbf{A}\in\mathbb{C}^{M\times N}$, show that $\mathcal{R}(\textbf{A})=\mathcal{R}(\textbf{AA}^H)$ where $\mathcal{R}()$ denotes the range space of a ...
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V-matrix in SVD of A

I have some doubts in understanding the concept of SVD. 1) Let's assume I have a matrix $$V^H= \begin{bmatrix} f &e&g&f\\ c &d&d&k\\ m &f&j&e\\ p &l&a&...
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Does the decay rate of singular values reflect linear dependency of vectors in a matrix?

In order to compare two matrices, suppose A and B, based on the level of linear dependency between column vectors of a given matrix, I can think about following measures: Rank of a matrix: If the ...
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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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Question about SVD proof from Trefethen and Bau

In Trefethen and Bau's proof of the SVD (see image below), they start by defining the following: $$ U_1^* A V_1 = \begin{bmatrix} \sigma_1 & w^* \\ 0 & B \end{bmatrix} $$ I understand the ...
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UV decomposition

I can't understand exactly what UV decomposition is and how it can be done on a square matrix. So, can anyone please explain me an example of UV decomposition on any square matrix say, a 4x4 or a 5x5 ...
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Regularised Linear least squares via SVD in Matlab

enter image description here Basically I'm trying to create a Matlab script where I can create the the Matrix S with diagonal entries as shown in equation 4 of the picture for different sizes n. ...
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Singular Value Decomposition of a Real Unit Matrix

Given a real matrix $A \in \mathbb{R}^{m \times n}$ whose entries are all ones, what is the reduced/short/economy singular value decomposition $A = U\Sigma V^T$? I can see that we have a single ...
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SVD - reconstruction from U,S,V

I am learning some linear algebra for image compression and I am stuck at this point: Suppose I have a matrix $R$, $$ \begin{bmatrix} 5 & 7\\ 2 & 1\\\end{bmatrix} $$ Then I compute ...
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What is the geometrical significance of a complex-valued singular value decomposition?

Suppose you have a 2x2 real-valued matrix, $\mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $\mathbf{A}$ into a 2-D ...
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Explain the geometrical meaning of Singular Value Decomposition (SVD)

Suppose you have a 2x2 real-valued matrix, $\mathbf{M}$. If you perform a singular value decomposition (SVD), then Wikipedia and the internet tell me that this can be understood geometrically as a ...
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How do I prove that singular values of A are the square roots of eigenvalues of $AA^T$

$A \in \mathbb{R}^{m\times n}$ My task is to show that the singular values of A are the square roots of the eigenvalues of $AA^T$ if $m \geq n$ or of $A^TA$ if $ m \leq n$ I'm able to show that ...
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Exercice base for SVD

I've been trying to show that Ker(K^T) is the same set as the orthogonal of Im(K) for SVD (Singular Value Decomposition) purposes. I did half of it as you can see in the image below, so can you help ...
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Find SVD of a matrix

Let A be a matrix, $A= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}$ and it's SVD, $A=USV^t$. Let $U^1$ be the first column of $U$ and $V^1$ the first ...
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Frobenius Norm Inequality with SVD

Let $A\in \mathbb{R}^{m\times n}$ and $x\in \mathbb{R}^n$ a column vector. I want to prove that $$||Ax||_2 \leq||A||_F||x||_2$$ using SVD where $||\cdot||_2$ is the euclidean norm and $||\cdot||_F$ ...
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Does the singular value decomposition coincide with the spectral decomposition for square matrices?

Assuming that $A$ is a diagonalizable matrix, does the singular value decomposition of $A$ coincide with its spectral decomposition? I think no, because in the spectral decomposition $A = Q^{-1} \...
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Multivariate normal distribution and sum of singular values

Using a multivariate normal distribution I've (numerically) computed the expectation value of $x^Tx$: $$ <x^Tx> = \frac{1}{\sqrt{(2\pi)^k|Z|)}}\int_{\mathbb{R}^k} x^Tx \exp(-\frac{1}{2}x^TZ^{-...
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can we perform SVD in spectral clustering to get top-$k$ eigenvector?

In spectral clustering, we need to compute top-k eigenvector for k-means clustering I have been told that SVD and eigendecomposition is equal for symmetric matrix here page 9. I try in matlab <...
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Relationships between top-$k$ eigenvector and top-$k$ singular vector of symmetric matrix $A$

Is there any relationships of top-$k$ eigenvector and singular vector of symmetric matrix $A \in R^{n \times n}$? For symmetric matrix $A$ its eigenvalue decomposition is: $$ A = B \Lambda B^T$$ ...
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Why eigenvectors of $\mathbf{A}^T\mathbf{A}$ are in row space of $\mathbf{A}$?

I'm following SVD proof and I can't get why eigenvectors of $\mathbf{A}^T\mathbf{A}$ are in rowspace of $\mathbf{A}$. I can understand further why these eigenvectors are basis for the row space of $\...
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Prove that rank(A) = rank(A$^\dagger$)=rank(AA$^\dagger$)=rank(A$^\dagger$A) using the SVD decomposition

Prove that rank(A) = rank(A$^\dagger$)=rank(AA$^\dagger$)=rank(A$^\dagger$A) using the SVD decomposition. $A^\dagger$ is a Moore-Penrose inverse of A. I managed to prove the first equation, $rank(A) ...
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Residual error of a normalized DLT

I use the Direct Linear Transform (DTL) to estimate a homography from a given set of point correspondences $(x_i,x_i')$, such that $$x_i' = \mathbf{H}x_i$$ Therefore, I set up the measurement matrix $\...
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1answer
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Column Orderings in any QR/Matrix Factorization Method

I am trying to understand if the ordering of columns matters in QR decompsoition. In general it seems that column ordering won't matter. I guess for SVD or any matrix factorization the way columns ...
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44 views

Condition for product of tow rectangular matirx is diagonalizable?

Let $A$ and $b$ be $m \times n$ matrices, it seems the product $A'B$ is diagonalizable only if $A$ and $B$ share the same left and right singularvectors. Is it true ?. How can I prove it, in case ...
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Using SVD to express the normalized norm of the output of a linear map

This is probably trivial, but it's been quite some time since my Linear Algebra exam, and my SVD skills are rusty. 1) Let $T$ be a linear map from $\mathbb{R}^n$ to $\mathbb{R}^n$, and $M$ the ...
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U matrix in Singular value decompositon.

I know that the Singular Value Decomposition of a matrix $X$ is given by: $X = U\Sigma V^T$, where $U$ and $V$ matrices are column orthonormal and $\Sigma$ is a diagonal square matrix containing ...
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Does SVD give the best rank 1 approximation with respect to the Frobenius norm, L2 norm, or both?

From what I've observed in practice the SVD gives the best rank 1 approximation with respect to the Frobenius norm. But from what I've heard from others, it also minimizes the distance to the L2 ...
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Kronecker product SVD Error bound

The Kronecker Product SVD (KPSVD) is defined here. Given a target rank $r$, what is the error bound in terms of singular values $\sigma_i$ for $\|A - A_r\|_F$, where $A_r = \sum_{i=1}^r \sigma_iU_i \...
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Low rank approximation of non-symmetrix square matrix

I have asked this in a previous post, and perhaps the question was not properly posed. I will try again. I have a square, real, non-symmetric matrix $A$ which satisfies $A + A^T \geq 0$. From $A$, I ...
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Eckart–Young–Mirsky theorem: rank $\le$ k or rank = k

The Eckart–Young–Mirsky theorem is stated sometimes with rank $\le$ k and sometimes with rank = k. Why? More specifically, why the following two optimization problems are equivalent: Given a matrix $...