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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Finding a maximal orthogonal basis from a set of functions

I have a set of functions $\{f_1,\ldots,f_n\}$ with an associated inner product $\langle f_j,f_k\rangle=\int d^2z f_1^*f_2$ . The functions are not linearly dependent; i.e. the rank $r<n$, where ...
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Compute a similarity matrix of documents from SVD (LSI)

I'm computing a SVD from a Matrix $m$ (columns = Documents ($D$) and rows = Terms ($T$) and truncate this matrix to lower the dimension of $m$ to $k$. From my resulting matrix $A = U \Sigma V^T$ I ...
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SVD - Decomposed Matrix Sizes

I had a question about SVD. Specifically about the size of matrices $U$, $\Sigma$ and $V$ decomposed from the $m\times n$ matrix $X$ using the formula $$X = U \Sigma V^T $$ Most of the the tutorial ...
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Find Singular value decomposition of Matrix

Find the Singular Value Decomposition of the $3\times 2$ matrix: $$ \begin{pmatrix} 1 & 1 \\ -1 & 1 \\ 1 & 1 \end{pmatrix} $$ My book says the first step is to find the vector $v_1$ and ...
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Singular value decomposition of the real part of a complex matrix

An $N\times N$ complex matrix $K$, with SVD of $K = U\Sigma V^H $. I only have access to the real part of the elements of $K$, i.e. $K_R$. If I did SVD of $K_R$ to get $U_R \Sigma_R V_R^H$, my ...
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characterization of the solution to a generalized eigenvalue problem

Let's say we have the following optimization problem. (All the $\Sigma_{ii}$'s are positive definite.) $\max u^\top \Sigma_{12} v\quad$ $\text{subject to}\quad u^\top \Sigma_{11} u = 1\quad and\quad ...
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Solving $Ax = 0$ with Singular Value Decomposition

In my computer vision class, we have seen the problem of solving $Ax = 0$ numerous times. Initially, we solved the problem by computing the smallest eigenvectors corresponding to the smallest ...
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Matrix norm inequality : $\| Ax\| \leq |\lambda| \|x\|$, proof verification

Suppose that $A$ is a normal $n\times n$ matrix. Show that $\|Ax \| \geq |\lambda_n|\|x\|$ for all $x \in \mathbf{C}^n$, if $\lambda_n$ is the eigenvalue to $A$ of smallest absolute value. Is ...
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Most efficient method for computing Singular Value Decomposition of a triangular matrix?

There are several methods available for computing SVD of a general matrix. I am interested in knowing about the best approach which could be used for computing SVD of an upper triangular matrix. ...
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Do similar matrices have equal singular values?

Is it true that if $A$ and $B$ are similar matrices, $B=S^{-1}AS$, then $A$ and $B$ have the same singular values?
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Singular Value Decomposition of $A=W^T V W$

I've given the following Matrix $A=W^{\dagger} V W$ where $\dagger$ denotes transpose+complex conjugation. It follows that $A$ is hermitian. $W$ is a $n \times m$ Matrix. $V$ is a diagonal $n\times ...
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Derivative of the singular values of the product of two matrices.

If $\mathbf{\Sigma}_{\mathbf{A}^H \mathbf{B}}$ is the diagonal matrix of singular values of $\mathbf{A}^H \mathbf{B}$, what is the derivative of ...
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SVD of a product of a Hermitian matrix and diagonals matrices

I have a Hermitian matrix $H$ and its SVD decomposition $H = V*S*V^T$. Let $D$ be a diagonal matrix. Is it possible to deduce the SVD of $D*H*D$ from the above SVD ?
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Understanding a derivation of the SVD

Here's an attempt to motivate the SVD. Let $A \in \mathbb R^{m \times n}$. It's natural to ask, in what direction does $A$ have the most "impact". In other words, for which unit vector $v$ is $\| A ...
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1answer
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SVD of a real symmetric matrix confusion.

For a real symmetric matrix A, it's true that: $ A A^T = A^T A = A^2$ And since the right and left singular vectors of $A$ are the eigenvectors of $A^T A$ and $A A^T$ respectively, the right and ...
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Relationship between eigenvectors of correlation and covariance matrices

For the purpose of computing principal components of a dataset, represented as matrix $X$ of dimensions $n \times p$ with $n$ samples and $p$ features, we can compute sample covariance matrix $S$, and ...
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1answer
60 views

why do we say SVD can handle singular matrx when doing least square? Comparison of SVD and QR decomposition

I don't quite understand why we say that QR decomposition doesn't handle singular matrix, while SVD does when they are used for least square problem? My example in Matlab seems to support the ...
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Higher order singular value decomposition

Can anyone give a clear explanation with example of the higher order singular value decomposition (HOSVD). All the references are theoretical stuff. I would like to have a simple example.
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SVD of Cholesky Factor

I am working through the book Fundamentals of Matrix Computations by David Watkins, and I ran into this one and it's stumping me. In my head, I understand the basic premise of it. However, I can't ...
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SVD Transpose Equations

$$Av_i= \begin{cases} \sigma_iu_i & i = 1, \ldots , r \\ 0 & i = r+1, \ldots , m \end{cases}$$ $$A^Tu_i= \begin{cases} \sigma_iv_i & i = 1, \ldots , r \\ 0 & i = r+1, \ldots , m ...
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Properties (range, null space, norm, rank) of RREF, SVD, and Reduced SVD of matrix A

Consider the following three decompositions of an n by m, rank-r, matrix A: RREF: A = LQ where Q is the RREF form of A and L is invertible. SVD: A=UΣV∗ where U and V are unitary and Σ only has ...
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Given matrices $A,B$, minimize $\|UAV^T - B\|_F$ over orthogonal matrices $U, V$

(Question edited to shorten and clarify it, see the history for the original) Suppose we are given two $n\times n$ matrices $A$ and $B$. I am interested in finding the closest matrix to $B$ that can ...
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Spectral Decomposition Theorem for Symmetric Matrices Converse

In my class, it was stated that any $n \times n$ symmetric matrix $\mathbf{X}$ may be written as $$\mathbf{X} = \mathbf{P}\boldsymbol{\Lambda}\mathbf{P}^{\prime}$$ where $\mathbf{P}\mathbf{P}^{\prime} ...
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$\underset{\dim(S) = k}{\max} \underset{y \in S}{\min} \frac{y^T (X^T A X) y}{y^Ty} \ge \underset{y \in S_0}{\min} \frac{y^T (X^T A X) y}{y^Ty}$?

Let $A \in \mathbb{R}^{n \times n}$ be symmetric and let $X \in \mathbb{R}^{n \times n}$ be nonsingular. Let $\lambda_k$ be the $k^{th}$ largest eigenvalue of $A$. Suppose for some $k$ we have that ...
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$y \in \mathbb{R}^n \implies \frac{y^T (X^TX) y}{y^Ty} \ge \sigma_n(X)^2$?

I came across an expression in a book on matrix computations that is not clear to me. Let $X \in \mathbb{R}^{n \times n}$ be nonsingular. Then we have $y \in \mathbb{R}^n \implies \frac{y^T (X^TX) ...
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How to determine $\ker(A^\mathsf{T})$ using the SVD of $A$

Imagine I have the SVD of $A$ so: $$A = USV^\mathsf{T}.$$ How can I determine the $\ker(A^\mathsf{T})$ from the SVD? I understand that the $\ker(A^\mathsf{T})$ consists of all vectors that are ...
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'Sign' of normalized eigenvector for singular value decomposition

I'm working on an SV decomposition script in Python. I am getting incorrect results because of the 'indeterminacy' associated with normalizing the singular vectors. I understand that the sign of the ...
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SVD of (2,1,-2) not ok

I'm trying to find the SVD of $$ \begin{pmatrix} 2&1&-2\\ \end{pmatrix} $$ I found $$\Sigma , u$$ But on the V matrix I got $$ \begin{pmatrix} ...
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Property of multiplation of a diagonal matrix

I have the singular value decomposition of an image: $X = U \Sigma V^T$, where $\Sigma$ is diagonal matrix. I want to reformulate this equation like this: $X=\Sigma D$ where $D$ is any combination of ...
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Finding the SVD for A given that A = QR and R = $U_r \Sigma V^T$

I'm studying for my final exam and one of the practice problems given to me by my professor asks me to write down the SVD for a matrix $A$ given the QR factorization of $A$ and the SVD of $R$. My ...
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Solution in terms of singular values and singular vectors

Given $(A^{T}A+L^{T}L)x=A^{T}b$ where $L=\alpha I$, I am asked to find $x$ in terms of $\alpha$ and singular values and vectors of $A$. If I try using $A=U\Sigma V^{T}$ I eventually get to ...
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sum of singular vector dyadics derived from the matrix itself

I have an m*n (m>n) n-rank matrix (let's denote it by A), with nonnegative elements. SVD decomposition says, that A=UDV', where U and V are orthogonal matrixes, and their columns are the singular ...
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eigenvectors in SVD decompositoon for pseudo inverse

I am usign SVD to calculate the pseudo inverse of a matrix. When calculating the SVD, I can choose the one of the eigenvectors of V as a relation of x to y of $x=-\frac{y}{9}$ Now, I can only get the ...
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How do I solve Wahba's problem in more than 3 dimensions?

The cost function that Wahba's problem seeks to minimise is as follows: $J(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k|| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k ||^2$ where $w_k, v_k$ are ...
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Interpretation of SVD for non-square matrices.

I was reading the Wikipedia article on Singular Value Decomposition. It shows a nice visualisation where the SVD of a matrix $M = U\Sigma V^*$ allows us interpret M as a rotation $V^*$, followed by a ...
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SVD for Seam Carving

Could SVD be used for Seam Carving ? I am making a small program for a uni course and I'm looking for different ways to calculate pixel energy; which made me come across SVD. Among others, I have ...
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Is it possible to have SVD decomposition with a square and diagonal middle matrix?

Suppose $\mathbf{H}=\mathbf{F\Lambda G}^*$ is the SVD decomposition, where $\mathbf{\Lambda}$ usually has the same dimension as $\mathbf{H}$. However, I am wondering whether it is possible to somehow ...
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Singular vectors of random Gaussian matrix

Let $A$ be a singular vector matrix of a random Gaussian matrix. The entries of the Gaussian matrix are i.i.d., so the singular vectors are distributed isotropically. Is it possible to get ...
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Computationally more efficient technique than Singular Value Decomposition.

I am working on a mathematical project where I have to decompose a given matrix into two or more matrices. Presently I am using Singular Value Decomposition (SVD) for it. I came to know that SVD is a ...
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Calculating Recommendations based on HOSVD

Two-dimensional case Given a user-movie matrix $\mathbf{M}$ that contains ratings, it can be decomposed using SVD to the product of $\mathbf{U}$, $\Sigma$ and $\mathbf{V^T}$. Now, given a new users ...
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SVD for augmented matrix

I'm doing the following problem to prepare for my exam tomorrow: Suppose that $A$ is a $n \times m$ matrix with $n \geq m$ and that the singular values of $A$ are $\sigma_1, \sigma_2, \dots, ...
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Null space for $\mathcal{N}(A)$ given SVD of $A$

Let $A \in \mathbb{R}^{m \times n}$ be a matrix with singular value decomposition $$ A = U \tilde{\Sigma} V^{T}. $$ Let $\text{rank}(A) = r \leq \min(m, n)$. My textbook notes that the first $r$ ...
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What are eigenvectors and eigenvalues in layman terms?

I am learning Singular Vector Decomposition (SVD) technique. It breaks a matrix X into 3 matrices U, S and $V^T$. U is formed by eigenvectors of matrix X. My understanding is that eigenvectors are ...
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Joint Problem: Solve offsetted Wahba's Problem

You may probably know Wahba's Problem, where an unknown rotation matrix $\mathbf{R}$ based on 2 vector sets in different frames is to be found: $$ \mathbf{b}_k = \mathbf{R} \mathbf{a}_k + ...
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spectral density of Guassian random matrix

I am interested in the spectral properties of Gaussian random matrix. I can see the constant dominance (mostly by the two most extreme ones-largest and smallest-) of the extreme eigenvalues in the ...
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Uniqueness of Singular Value Decomposition

There is a theorem that states that if A is a square matrix and the singular values are distinct, the left and right singular vectors are uniquely determined up to complex signs. Can someone give me ...
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SVD of a Matrix Product

Suppose we have a matrix $A$ with dimensions $m$ by $n$ and a column-wise permutation matrix $R$ (re-orders columns) with dimensions $n$ by $n$. Then we have a matrix $X$ which is constructed as $X ...
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Identifying linearly dependent columns of a matrix

I am using SVD to fit a polynomial surface to a set of points, where the number of points may be less than, equal to, or more than the number of polynomial terms. For simplicity, let's assume points ...
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29 views

SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
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Null Bilinear Forms $x^T A y = 0$, where $A$ is square and full rank.

Let A be a full rank square matrix (A has no null space). When does $y^T A x = 0$ occur ? It could be that this problem is case-specific, so please find attached a document where x,y, and A take ...