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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How do we know V in the SVD is the eigenvectors of M*M?

if we have any real matrix M nXm, the SVD (singular value decomposition) allows us to decompose it into $U{\Sigma}V^T$, where V is an orthogonal real matrix composed of the eigenvectors of $M^TM$. ...
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Proper Generalized Decomposition

I have researched some topics regarding Proper Generalized Decomposition. All the text more or less state the what is the algorithms to conduct PGD as Assume that the solution is separable in x(space)...
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Trying to understand why the largest singular value of $M_n^{-1}$ is $1.4286$ no matter what $n$ I choose

Let $n \ge 3$ be an odd number. Let $I_n$ be the $n$ dimensional identity matrix and let $A_n$ be the $n\times n$ matrix where every element is zero except the central element which is, say, $0.3$. ...
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How to find the SVD when eigenvalue is $0$? [duplicate]

After calculating the eigenvalues, I get $1040400$ and $0$. Since one of them is $0$, how do I calculate (orthogonal) matrix $U$? $$u_i = \frac{1}{\sqrt{\lambda_i}} A v_i$$ I know there are a similar ...
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Proof that A+A projects onto R(A) and AA+ projects onto N(A)

I have been able to understand the proof that shows both are projections by proving P^2 = P for both of them. I don't understand how these projections project onto R(A) and C(A) though. Proof that ...
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Can TDT^T transform a square matrix D into diag{D_0,0} where D_0 is nonsingular?

If $D\in R^{n\times n} $ is a square matrix with rank $m$ ($m < n$), can we always find a nonsingular matrix $T\in R^{n\times n}$ such that $$ TDT^T = \left[\begin{matrix}D_0 & 0\\ 0& 0\end{...
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Singular Value Decomposition Theorem: Corollary

In a linear algebra and analysis course [it's a hybrid course between the two], we recently had the SVD (singular value decomposition) theorem, and the prof. told us (due to lack of time without proof)...
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How to calculate singular vector for symmetric matrix from eigenvectors

For some reason, I need to calculate singular vectors of a symmetric matrix from its eigenvectors. I think they are the same subject to the signs of eigenvalues. I know that someone uses QR ...
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About condition number

I have the following exercise: Relate the 2-norm condition of $X\in \Bbb R^{m\times n}\ (m\geq n)$ to the 2-norm condition of the matrices: $$B=\begin{equation} \begin{bmatrix} I_m & X\\ 0 & ...
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singular values of $A-\alpha I_n$ while $A\in\mathbb{C}^{n\times n}$

I have to prove that singular values of $A-\alpha I_n$ are $\sigma_i+\alpha$, while $\sigma_i$s are singular values of $A$ and $A$ is hermitian and positive definite matrix. Also we know that: $$\...
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Matrix whose columns are the first $p$ left singular vectors [migrated]

Let's say I have a $m \times n$ matrix $A$. When I find its SVD, I get $p$ dominant singular values. How do I get the $m \times p$ matrix whose columns are the first $p$ left singular vectors of $A$?...
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The optimal solution of tensor Tucker decomposition problem

I have a question about tensor Tucker deocomposition. Recall that in Higher-Order Singular Value Decomposition (HOSVD), each factor matrices $U_i$ is obtained by taking the top $R_i$ left singular ...
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Fast computation of singular values of multiple matrix products with a diagonal matrix

Suppose $A \in \mathbb{R}^{n \times m}$ is a rectangular matrix and $D_i \in \mathbb{R}^{m \times m}$ for $i = 1, \ldots, N$ are diagonal matrices. I'm interested if there exists a fast way to compute ...
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How is SVD better than Gaussian elimination in finding the rank of a matrix?

In Linear Algebra and its Applications, Gilbert Strang, $4^{th}$ ed, one of the applications of SVD is mentioned as finding the effective rank of a matrix. The idea presented in the book is that the ...
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Matrix reconstruction puzzle

Say a reconstruction of matrix $A$ is $A'$ and it's defined as $ A' = P^TDPA $ where $P$ is an orthonormal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal ...
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How to make a multidimensional SVD?

Is it possible to define a tensor Singular Value Decomposition (SVD)? For example for the 3 tensor $$\left[\begin{array}{rr}1&-1\\1&1\\\hline 1&-1\\-1&-1\end{array}\right]$$ Can be ...
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For an $n \times n$ matrix whose SVD is $A = U D V^\top$, what do we know about $UV$?

I'm using a svd (singular value decomposition) function in a programming library that I didn't write. Given a square real nxn matrix, svd returns three values U,S,V where S is a vector designating a ...
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Should the svd of a square matrix always compute a U and V which are inverses?

I’m using a Scala library which (I believe) wraps a java library which does some linear algebra computations. There is a function which is documented to return THE singular value decomposition. First ...
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Maximizing correct(or incorrect) labeling and prove its bound.

Suppose there are M workers labeling N items (items have true label +1/-1). For each worker, it has $\frac{1+p_i}{2}$ $(-1\le p_i\le 1)$ successful probability to recover the true label of an item. ...
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Solving system of equations when unknowns are multiplied with each other

I am trying to solve a system of linear equations related to camera calibration as provided in the paper https://downloads.hindawi.com/journals/mpe/2016/1392832.pdf. I am trying to solve Equation ...
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Find spectral norm via power iteration

I have a question about this algorithm to find the spectral norm via power iteration from the paper Spectral Normalization for Generative Adversarial Networks. I don't know if I understood it ...
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Singular Value Decomposition and lower-rank projection

Suppose we have matrices $Q, K, V \in \mathbb{R}^{n \times n}$ and $P = QK^\top$ is of rank $k$. We want to prove that there exist two matrices $C, D \in \mathbb{R}^{k \times n}$ such that $PV = Q(CK)^...
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SVD operational interpretation

I was reading a textbook discussing the SVD's interpretation and it states the following: Consider a matrix A mxn whose SVD is given by A=U\SigmaV^{T} Then "the r orthonormal basis vectors for N(...
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Singular Values of a Toeplitz Matrix

I am looking for analytical expressions for the the singular values of a Toeplitz matrix. If possible for a general Toeplitz matrix but I would also take results for a tridiagonal Toeplitz matrix \...
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Simulate multivariate normal when covariance matrix is not positive semi definite and has negative eigenvalues

Given a matrix $\Omega$ of size $m \times m$ that is not positive semi definite (some of the eigenvalues are negative) but symmetrical, I want to simulate $\pmb{Z} \sim N_{(m)}(\pmb{0}, \Omega)$. I ...
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Questions about SVD, Singular Value Decomposition

I am not a mathematician, so I need to understand what SVD does and WHY more than how it works exactly from the math perspective. (I understand at least what is the decomposition though). This guy on ...
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Show that every rank-deficient matrix has a full rank matrix arbitrarily close to it

I came across the following exercise in preparation for a test. Let $A \in \mathbb{R}^{n \times m}$ with rank $r < \min\{n,m\}$. Use SVD of $A$ to show that for every $\epsilon >0$ (no matter ...
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Why is the does the A here equals this summation?

In an explanation of the SVD for hilbert space, hence the bra-ket notation, the matrix A is simplified as this summation below. Why is this the case?
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Singular values of “over-full” rank matrix

Singular value decomposition of a matrix $\mathbf{A}\in\mathbb{C}^{r\times r}$ can be written as $$\mathbf{A}=\sum_{i=1}^{r}\sigma_i\mathbf{u}_i\mathbf{v}_i^H$$ However, I have a matrix $\mathbf{B}\in\...
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Matrix Squareroot Using Single Value Decomposition (SVD)

I'm trying to find the square root of a square matrix(real or complex) that can have complex eigenvalues and eigenvalues. The approach that I'm using requires me to first calculate the ...
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Solving for $U$,$\Sigma$ and $V$ in SVD of a matrix

Knowing from theory that fro a matrix $A$ we $A = U\Sigma V^{T}$ I want to solve for $U$,$V$ and $\Sigma$. My effort is the following but I don't if I am correct. \begin{align*} A &= U\Sigma V^{T} ...
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CP-ALS algorithm with Kroonenberg and De Leeuw initialization

I am trying to implement the CANDECOMP/PARAFAC - Alternating Least Square algorithm, with a small improvement in the initial guess. The simple initial guess for A is random in the method==0 if block. ...
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SVD in scipy and numpy for tensors

Can someone explain to me the difference between SVD of numpy and scipy for Multidimensional arrays (Tensors)? ...
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why use svd() to invert a Hermitian matrix?

In MATLAB, I compared elapsed time to invert a Hermitian matrix using inverse(), svd(), and ...
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Inequality of singular value of product of matrices

If $P$ and $Q$ are $n$-sized square matrices over the complex field, and $ \sigma_1(C) \geq \cdots \geq \sigma_n(C)$ denote the singular values of $C$ then how to prove that $\sigma_1(PQ) \geq \text{...
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How does a column of zeros in a matrix influence its right singular vectors?

Here is the SVD decomposition calculated in Julia: Why is the 4th column of $V^{T}$ also zero? How is that connected with columns of $A$? Appreciate any help!
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Finding the (condensed/reduced) SVD of a given matrix by inspection

$ \newcommand{\tp}{^\mathsf{T}} \newcommand{\R}{\mathbb{R}} \newcommand{\RR}[2]{\mathbb{R}^{#1 \times #2}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\m}[1]...
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Why is $||Ax||=||x||$ if and only if $A^TA$ is the identity matrix?

I am going through my lecture nots in linear algebra and during a proof we state that $||{Ax}||_2^2=<Ax, Ax>=<x, A^TAx>=<x,x>=||x||_2^2$ if and only if $A^TA=E$ with $E$ being the ...
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A strange characterization of the minimum singular value of a matrix

Let $A = \begin{bmatrix}a_1 | \cdots|a_n \end{bmatrix} \in\mathbb{R}^{n\times n}$. Is it true that $$\sigma_n(A) = \min_{1\leq j\leq n} \min_{\alpha\in\mathbb{R}^{n}}\|a_j - \sum_{i=1,i\neq j}^{n}\...
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Condensed SVD decomposition of an outer product

Let $A = uv^{T} \in \mathbb{R}^{m \times n}$. Find the (condensed) SVD decomposition of $A$. Theorem (Condensed SVD decomposition) Let $A \in \mathbb{R}^{n \times m}$ be a non-zero matrix of rank $r$....
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Subgradients of the trace norm for a singular value decomposition

I'm following CMU convex optimization course. I am doing homework 2 but I'm stuck at this question:$\DeclareMathOperator{\tr}{tr}$ For $f(X)=\lvert\lvert X \rvert\rvert_{\tr}$, show that the ...
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Finding Orthonormal Basis using SVD and comparing it with Gram-Schmidt shows different result

I was trying to find the orthonormal basis for the column space of the following matrix "A" \begin{pmatrix} -1 & -1 & 2 & 3 \\ -1 & 1 & -3 & -4 \\ 2 & -2 & 5 ...
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Find SVD of the given matrix

Find a decomposition $X=U \Sigma V^{T}$ of the matrix $X=\begin{bmatrix} 2 & 1 & 2\\ -2 & -1 & -2\\ 4 & 2 & 4\\ 2 & 1 & 2 \end{bmatrix}$ where $\Sigma$ is a ...
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Find a decomposition of the given matrix

I have encountered a task, which I do not really know how to approach. I need to find a decomposition $$A = U\Sigma V^T$$ of the matrix: $$ A = \begin{pmatrix} 2&1&2&\\ -2&-1&-2&...
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How to show that each rank one decomposition of SVD is exactly rank one?

by using the fact that rank(AB) $\leq$ min{rank(A), rank(B)}, I can only figure out that rank(u$\cdot$v$^{T}$) $\leq$ 1. How to show that it is exactly rank one? Appreciate any help!
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Conditions for equality for the nearest rank-$k$ matrix

my proof for the best rank approximation, using the Frobenius norm, is as follows: \begin{equation} \begin{alignedat}{1} \vert\vert{\mathbf{A-B}}\vert\vert_F^2 &= <\mathbf{A - B}, \mathbf{...
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Multiplying an eigenvector by $-1$ while constructing the V matrix in SVD decomposition

While performing SVD I found eigenvectors, which will allow me to write down my $V$ matrix. However, I need to multiply the 3rd eigenvector by $-1$ because it will satisfy some condition that my task ...
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Swapping the orthonormalized eigenvectors in the V matrix while performing SVD decomposition.

While performing an SVD decomposition, I found eigenvalues and eigenvectors, I orthonormalized them and they look something like this: $$\mathbf{eig_{1}}=\left(\begin{array}{c} a\\ b\\ c \end{array}\...
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Rank one decomposition of a positive semi-definite matrix with inequality trace constraints

Suppose there is a square matrix $A$ and a positive semi-definite matrix $X\in\Re^n$, such that \begin{equation} \mathrm{trace}(AX)\leq0 \end{equation} Is there any ways I could do the rank one ...

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