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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Orthogonality of left and right singular vectors of traceless 2D matrices

Let $A$ be a traceless $2\times 2$ complex matrix. Its SVD reads $A=UDV^\dagger$, or in dyadic notation, $$A=s_1 u_1 v_1^\dagger+s_2 u_2 v_2^\dagger,$$ with $\langle u_i,u_j\rangle=\langle v_i,v_j\...
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SVD - Finding the angle of rotation from U and V

Given a 2×3 matrix, the Singular Value Decomposition would give the matrix U which would be a 2x2 matrix and VT (transpose of V), a 3x3 matrix. From what I ...
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Dimensionality Reduction Using Low Rank Approximation

The problem: Given a sequence $\left \{ x_i \right \}_{i=1}^N \subseteq \mathbb R^n$ we want to find the best "compression" of these vectors onto a $p$ dimensional affine space. This means, I want to ...
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35 views

What units do singular values of a matrix have?

Consider a matrix A (containing elements that possess N as unit) that maps a vector b (containing elements that possess m/N as unit) to a vector c (containing elements that possess m as unit). What ...
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Let A be a 3x3 matrix with singular values 3,4,5. How many SVD decomposition are there? [closed]

There are 3! ways of ordering the singular values. So there is 6 different SVD. Am I right?
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When mean corrected matrix $X_c = UΛV^T$, how to use this singular value decomposition to prove its three spatial properties?

We use the singular value decomposition on a mean corrected data matrix. $X_c = \begin{bmatrix}(x_1-x̄)^T\\(x_2-x̄)^T\\⋮\\(x_n-x̄)^T\end{bmatrix} = UΛV^T$, Let $\sqrt{n-1}U = (\dfrac{x_c\hat{e_1}}{\...
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Proof Help : $A \in \mathbb{R}^{n \times n}, \sigma > 0$ if and only if the next matrix is singular.

I've been stuck with the next proof. I've tried to approach the first implication using that if $\sigma$ is singular value of A, then is solution of $ P(\lambda) = \det(A^t A - \lambda I)$, but i don'...
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SVD with rectangular matrix U

I have a matrix $A_{3\times2}$ , $U_{3\times2}$ , $V_{2\times2}$. I was asked to find the singular values, which I did with by multiplying $U'AV$. I got the $U'$ by following this rule: Since $U$ is ...
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How is SVD computing unitary square matrices for rank-1 matrices (Matlab)

Consider matrix $\mathbf X=[\mathbf x ~\mathbf x] \in \mathbb R^{D \times 2}$. Of course, $\mathbf X$ has rank-1. Background: $\bullet$The full Singular Value Decomposition (SVD) of $\mathbf X$ is ...
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If $(U,\Sigma,V)$ is a singular value decomposition of $A$, do the first $\text{rank}A$ columns of $V$ and $U$ form orthonormal bases?

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ and $|A|:=\sqrt{A^TA}$ $r:=\operatorname{rank}A$ $\sigma_1>\cdots>\sigma_r>\sigma_{r+1}=\cdots=\sigma_n=0$ denote the singular values of $A$ ...
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Truncated singular value decomposition and error matrix

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ $r:=\operatorname{rank}A$ $\sigma_1>\cdots>\sigma_r>\sigma_{r+1}=\cdots=\sigma_n=0$ denote the singular values of $A$ We say that $(U,\...
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Strange SVD Bound with Frobenius Norm

For any matrix $A$, show that $$ \sigma_k \le ||A||_F/\sqrt{k} $$ where $\sigma_k$ is the $k$-th singular value of $A$. For $k=1$ I would say it's trivial, but for $k>1$? Also tried this looking ...
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Solving a linear matrix equation with both left and right multiplication of unknown

I would like to solve a matrix equation of the form $$ \mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{A}^T = \mathbf{B} $$ where $\mathbf{A}$ and $\mathbf{B}$ are known $n \times n$ matrices, and $\...
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Solving for $X$ using the SVD of $QX$ when $Q$ is orthogonal

I inherited some code (see below), and I am not quite sure what it does. It is part of a factor analysis-type model that learns a latent variable $X \in \mathbb{R}^{N \times K}$ with $N > K$ that ...
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If the singular value decomposition $U\Sigma V^T$ has rank $r$ and $Q$ is semi-orthogonal, then $Q\Sigma V^T$ has rank $r$ as well

Let $m,n\in\mathbb N$ and $A\in\mathbb R^{m\times n}$ with reduced singular value decomposition $U\Sigma V^T$, \begin{align}U&\in\mathbb R^{m\times r},\\\Sigma&\in\mathbb R^{r\times r},\\V&...
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SVD of a 3x3 symmetric matrix shortcuts

Say there is a 3x3 symmetric matrix such as the following: $$A=\begin{bmatrix} a & b & a \\b & b & b \\ a & b & a \end{bmatrix}$$ By being symmetric:$$AA^T=A^TA=A\cdot A =A^2$$...
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Why does SVD solve $\underset{U,V}{\min}\| A - UV^T\|_F^2$

I read here the following: You can solve the quadratic problem below through Singular Value Decomposition (SVD) of the matrix $A$. \begin{align} \underset{U,V}{\min} \| A - UV^T\|_F^2 \end{...
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The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$

Problem: Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that $$\|S-I \|_F \leq \|S-Q\|_F$$ for all orthogonal matrices $Q$, where $I$ is ...
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Relation between eigenvectors and singular vectors of complex skew-symmetric matrices

As shown in this answer, if $A$ is a real skew-symmetric matrix, and $v,w$ are a pair of orthogonal singular vectors with $$Av=sw \qquad\text{ and }\qquad Aw=-sv,$$ for some $s>0$, then the ...
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In SVD why is $\Sigma$ the square root of $V$'s Eigen values?

Following a problem, it was not explained why the $\Sigma$ matrix is the square root of $V$'s Eigen values rather than the values themselves.
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How do you compute the reduced SVD?

I know how to compute the full SVD by hand, but surprisingly, I couldn't find much information on how to compute the reduced SVD by hand. What is the easiest way to do this?
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Finding the Null Space Error

I am watching a video on SVD where the instructor is solving the the nullspace of $$\begin{bmatrix}26 & 18\\\ 18 & 74\end{bmatrix}$$ wich has the Eigen values $\lambda_1 = 20$ and $\lambda_2 =...
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Is range of A equal to range of AB?

I have a simple question. Actually I just tried to solve the question 'Is range of $A$ equal to range of $AA^TA$'. But it looks like much general question to ask 'Is range of $A$ equal to range of $...
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What are the most relevant applications of polar decomposition?

Assume there exists a new and very efficient algorithm for calculating the polar decomposition of a matrix $A=UP$, where $U$ is a unitary matrix and $P$ is a positive-semidefinite Hermitian matrix. ...
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SVD - why do we complete matrices $U$ and $V$ with the vectors that form a basis for the nullspace?

SVD picture Normally, in theoretical demonstrations, $U$ has size $(m, n)$ instead of $(m, r)$. Why is that so? Is there a reason why we include a basis for the nullspace as well in the matrices $U$ ...
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SVD for discovering patterns?

I have a matrix of 1000 rows as the instances or observations of some kind, values are between 0-1. Every row has 10 positions as columns, 1000 rows X 10 columns. The data is outliers-free. Every row ...
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Distribution of the entries of a particular matrix product

Let us assume a Complex Gaussian i.i.d. matrix $A$ which can be decomposed using the SVD into $A=UDV^*$, where $U$ and $V$ contain the left and right singular vectors, respectively, and $D$ is a ...
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Proof that transposed matrix can be used for ZCA

I am using Zero-phase Component Analysis (ZCA) for the processing of images. The images are provided as a matrix $X$ with $m=$ number of rows = number of images and $n=$ number of features (pixels) = ...
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SVD convergence on eigenvectors

I’m trying to understand why QR iteration results in convergence of the SVD matrices on eigenvectors and eigenvalues rather than arbitrary similar matrices?
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diagonal form of a complex matrix

Let us consider a complex symmetric matrix \begin{equation} A = \begin{pmatrix} a+ib & c \\ c & -a+ib \end{pmatrix} \end{equation} where the coefficients $a,b,c$ are real. I am interested in ...
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Singular value decomposition.

I would like to solve the following problem with singular value decomposition. We have $x_{1},...,x_{m}$ vectors of $\mathbb{R}^{d}$. But $d$ is too high for your computer. So Let's find a linear ...
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54 views

Find $\text{min}{F_{D}(X)}$ for an arbitrary diagonal matrix

Let $\mathbf{D}$ be a diagonal matrix of size $n\times n$ defined over a field $\mathbb{R}$, and $\mathbf{X}$ be a matrix of $n \times n$ for which $\text{rank}(\mathbf{X}) = 1$. Let $$F_\mathbf{D}(\...
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Singular values of $A$ obtained as square roots of eigenvalues of $AA^T$ : find matching singular vectors?

I'm studying the implementation of several SVD algorithms. Some of them want a symmetric matrix, so you can decompose $AA^T$ or $A^{T}A$ and get the singular values of $A$ as square roots of the ...
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Find SVD of $A (A^T A)^{-1} A^T$

This is exercise 4.2.12 of Fundamentals of matrix Computations from Watkins: Let $A \in\mathbb R^{n\times m}$, $n \geq m$, $\operatorname{rank}(A) = m$ with complete SVD $ A = U\Sigma V^T$, $U\in \...
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How to find proper SVD components?

My approach is based on this method: we want to find $U$, $\Sigma$, $V$ so that $A = U \Sigma V$. Then $A^T A = V^T \Sigma U^T U \Sigma V$. Since $U$ is an orthogonal matrix, this equals $V^T \Sigma^...
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Fundamentals of Matrix Computations, Watkins, exercise $4.3.9(e)$, SVD.

Given that $$A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6\end{bmatrix}, \qquad b=\begin{bmatrix} 1 \\ 1\\ 1\end{bmatrix},$$ what is the method to find all solutions of the least-squares ...
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Singular value decomposition (SVD) for non-symmetric square real matrix contradicts spectral theorem?

From Bretscher's Linear Algebra with Applications: where $A$ is a real matrix in $ \mathbb{R}^{n \times m}$ and the singular values of $A$ are the square roots of the eigenvalues of the symmetric $A^...
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Singular value decomposition in the language of operator theory

Let $H_i$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H_1,H_2)$ be compact, $|A|:=\sqrt{A^\ast A}$ and $\sigma\in\mathbb R$. How would we describe the singular value decomposition of $A$ in ...
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Proof of SVD: $Ax = U\Sigma V^Tx$

Suppose we have a matrix $A \in \mathbb{R}^{n \times d}$ whose SVD is $A = U\Sigma V^T$. We only have the condition that $AA^T = U\Sigma^2 U^T$ and $A^TA = V\Sigma^2 V^T$, and we cannot make any other ...
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How to find the condition number of a 3*3 matrix with singular value 10, 1 and 1/10? [closed]

If the singular values of a 3*3 matrix are 10, 1 and 1/10 respectively, what is the condition number of the matrix? Is such a matrix numerically tractable?
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Looking for a simple explanation of Singular Value Decomposition in practice

tl/dr: I'm trying to find the best rotation between two 3d point clouds, and all the answers say "use SVD", but I don't have the math background. However, once I get the concept, hopefully I can use ...
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Property of the projection onto Singular Vectors

I come across this property while reading a paper and cannot find it anywhere: The property goes as follow: given a matrix $A \in R^{N\times n}$, denote $ U,V $ as the subspaces spanned by left and ...
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Compute rotation matrix from fitting a plane with its own coordinate system

I'm trying to find rotation matrix for a plane from given 4 points $X_1 ... X_4$ that form a square. These points are fixed on a rigid square that can take any position in world coordinate system. I ...
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39 views

maximal singular value inequality

Given any real-valued, symmetric, positive semi-definite matrix $A, B \in \mathbb{R}^{n \times n}$, $\lambda > 0$. Does the following inequality hold for some constant $c$? $$\sigma_{\max}\Big((A+B ...
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Is gaining enough understanding of SVD/QR factorization until you have their formulas memorized useful?

Is knowing how to compute SVD/QRfactorization,Power iteration/power method by hand without notes useful in applications such as Statistics? Now adays, we can do everytime by Mathematica/computer ...
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SVD for image compression

I want to make sure I understood the concept behind SVD for image compression. So, we start off with a rectangular $m \times n$ matrix that stores all the pixel values of the image. We then compute ...
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Determine the vector that produces the biggest deviation

Given a set of points $a_i(x,y)$ in the 2-Dimensional plane how can I determine the unit vector $u$ which produces the biggest standard-deviation in the $<a_i,u>$ (scalar product) values? I ...
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If $A_n$ is a compact operator, determine the singular values of the product $A_n\cdots A_1$

Let $H$ be a $\mathbb R$-Hilbert space and $A_n\in\mathfrak L(H)$ be compact for $n\in\mathbb N$. How can we determine the singular values of the product $B_n:=A_n\cdots A_1$? We know that $$A_n^\...
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Economic form of the singular value decomposition

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ and $|A|:=\sqrt{A^TA}$ $r:=\operatorname{rank}A$ $\sigma_1,\ldots,\sigma_r$ denote the singular values of $A$ and $\sigma_i:=0$ for $i\in\{r+1,\ldots,...
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22 views

Two-sided Jacobi SVD vs One-sided Jacobi SVD

In both algorithms we're applying a sequence of Jacobi rotations $J_n$ to implicitly diagonalize $A^TA:$ $$J_n^T...J_1^TA^TAJ_1...J_n=\Sigma \iff (AJ_1...J_n)^T(AJ_1...J_n) =\Sigma \iff (AV)^T AV = \...

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