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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SVD approximation, construction of the matrix A such that A=USV*

I want to construct a matrix $\textbf{A} \in \mathbb{F}^{m \times n}$ such that $\textbf{A} = \textbf{US}\textbf{V}^{\ast}$. My goal is to do reverse SVD after I select the desired singular values for ...
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Confused about SVD, why does it not imply all matricies are diagonal?

For SVD decomposition, if $X = U \Sigma V^T$. Then for $$XX^T=U\Sigma V^T V\Sigma U^T=U \Sigma I \Sigma U^T = U \Sigma^2 U^T = \Sigma^2 \ ?$$ I believe $U$ and $V$ are both orthonormal matrices so ...
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Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A=R^* R$ is a Cholesky factorization of $A$

Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A = R^* R$ is a Cholesky factorization of $A$. In my book it says that I should use the Singular Value Decomposition. I have that $\rho(A)=\sqrt{\rho(A^*A)}=...
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SVD [Singular Value Decomposition] on Transformation Matrix

svd(T) = u sigma v Here I understand meaning of each and every term and why SVD is important. But I am failing to interpret this equation from Linear Algebra ...
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Condition number — what is it? [closed]

May someone tell me what the condition number of a matrix is and how it's related to SVD decomposition? I google a lot but got zero results on this.
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What’s an intuitive explanation of the SVD matrix factorization formula?

So I initially learned about this topic around a year ago in my second semester of linear algebra, but try as I might, I could never figure out how the heck the formula does what’s it supposed to do. ...
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approximation of matrix contains parts of 3mode vectors

Can someone explain to me what means the following statement: K-rank approximation of a matrix that contains parts of 3-mode vectors
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Singular value decomposition of product involving orthogonal matrices

Suppose that $D\in\mathbb{R}^{m\times m}$ is orthogonal, $\Sigma \in \mathbb{R}^{m\times n}$ only has elements on the main diagonal, and $V\in\mathbb{R}^{n\times n}$ is orthogonal, with $n < m$. ...
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Quadratic matrix bounds

Let A be a singular matrix with a simple (non-repeated) zero-eigenvalue. Dose the following inequality hold? $$\|Ax\|^2\geq\sigma_2\|x\|^2, \qquad \forall x\notin Null(A)$$ where $\sigma_2$ is the ...
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Given positive definite $X\in\mathbb{R}^{4\times 4}$, find $Y\in\mathbb{R}^{4\times 2}$, such that $YY'\approx X$

Given positive definite $X\in\mathbb{R}^{4\times 4}$, I want to find $Y\in\mathbb{R}^{4\times 2}$, such that $YY'\approx X$. My attempt: Using SVD, $X=U\Sigma U^*$. Let $U_i$ be $i'$th column of $U$,...
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Pseudoinverse and SVD

For the SVD $$\textbf{A=U}\boldsymbol\Sigma \textbf{V}^{*}$$ Where $\textbf{U}$ and $\textbf{V}$ are unitary By partitioning the matrix $\textbf{A}$, we have the following: $$\textbf{A} = \left[\begin{...
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Given $\sum_i\sum_j \sigma_{ij} a_i b_j^T$ and orthogonal $\{a_i\}$, find orthogonal $\{b_j\}$

Suppose I have a matrix $T \in \mathbb{R}^{n \times n}$ of the form: $$T = \sum_{i = 1}^{n_A} \sum_{j = 1}^{n_B} \sigma_{ij} a_i b_j^T$$ where $1 \leq n_A, n_B \leq n$, $\sigma_{ij} \in \mathbb{R}\...
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interpretation of an operation using singular value decomposition

Say I have a frequency response of a $2 \times 2$ system at a fixed frequency: $G(i \omega$). It can be anything, for example $\begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}$. If I compute the ...
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relation of singular values of A and A+E with spectral norm of E

let $A$ an $n\times m$ matrix and $p = \min(m, n)$. if $\{\sigma_1 ,\sigma_2,...,\sigma_p\}$ and $\{\alpha_1,\alpha_2,...,\alpha_p\}$ be the all singular values of $A$ and $A+E$ respectively, the ...
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Can we extend the difference between the singular values to a rank one convex function?

Let $M_2$ be the space of real $2 \times 2$ matrices, and let $\text{GL}_2^+ \subseteq M_2$ be the group of matrices with positive determinant. Define $f:\text{GL}_2^+ \to \mathbb R$ by $$ f(A)=|\...
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minimum singular value for bigger matrix and small matrix

Suppose $r\ll p$, and $A_1,A_2,A_3 \in \mathbf{R}^{rxp}, rank(A_i)=r$. Define minimum singular value as $$\sigma_r(A_i)=\text{r-th singular value of} A_i$$ also define $\kappa_{\min }=\min \left\{\...
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Proof of Orthogonal Procrustes

Having the equation where both R and Upsilon are symmetric matrices \begin{equation} R^{T}\Upsilon = \Upsilon^{T}R \end{equation} If we use the singular value decomposition of Upsilon, it can be ...
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If a convex combination of conformal matrices is conformal, are they all proportional?

$\newcommand{\CO}{\text{CO}}$ $\newcommand{\SO}{\text{SO}}$ $\newcommand{\dist}{\text{dist}}$ Let $\CO(2) =\{\lambda R : R \in \SO(2)\, | \, \lambda > 0\} $ be the set of $2 \times 2$ conformal ...
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Inequality related to SVD first singular value

I am trying to solve the following problem: Let z $\in \mathbb{R}^{n}$, A $\in \mathbb{R}^{m \times n}$, $m \geq n$ and let $B = \begin{pmatrix} A \\ z^T \end{pmatrix}$. Let us call $\sigma_1(C)$ to ...
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SVD of A with orthogonal columns

I am trying to solve the following linear algebra problem: Suppose that A $\in \mathbb{R}^{m\times n}$ has orthogonal columns $w_1,...,w_n$ where $\| w_i\|_2 = \alpha_i > 0$ Find the matrices $U, \...
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How to prove that $U$ and $V$ are orthogonal?. If, $MM^TU=U\Sigma^2$ and $M^TMV=V\Sigma^2$ then

Let $M \in R^{m×n}$. In the context of the SVD theory, how to prove that $U$ and $V$ are orthogonal , If $U$ and $V$ are invertible and, $MM^TU=U\Sigma^2$, $M^TMV=V\Sigma^2$. Orthogonality definition:...
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How to prove that if $MM^TU=U\Sigma^2$ and $M^TMV=V\Sigma^2$ then $U$ and $V$ are orthogonal?

In the context of Singular Value Decomposition. Suppose $U$ and $V$ are invertible. How to prove that if $$MM^TU=U\Sigma^2$$ and $$M^TMV=V\Sigma^2$$ then $U$ and $V$ are orthogonal? In the context of ...
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44 views

A beginner’s explanation for PCA on a multivariate time series

This is very much a beginner’s question. Say you have a 10 dimensional vector for every day in a time series of 100 days. I was reading about using PCA to reduce this to a one dimensional time series....
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Comparing similar vectors and detecting expressions in n-dimensional space

I have more of a methodological problem than an exact mathematical formula to solve. At first I'll describe the environment: I have a large set of delay-vectors (e.g. for a chip/cell) characterized ...
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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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350 views

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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37 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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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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29 views

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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24 views

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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