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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Singular value decomposition of a matrix and its transpose

Consider a matrix A. The SVD of A will be : A =USV'. The SVD of A' will be: A' = U1*S1*V1' (say). Since, A' = V*S'*U', therefore the values of U1, S1, and V1 should be as follow: ...
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Doubt about SVD algorithm

I am studing the SSA (Singular Spectrum Analysis) algorithm and after reading some papers about SSA, the ideia was not consolidated. I found a Matlab SSA algorithm here and I was trying to implement ...
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54 views

Decompose a matrix into diagonal term and low-rank approximation

For a matrix $A$ the Singular Values Decomposition allows getting the closest low-rank approximation $$A_K=\sum_i^K\sigma_i \vec{v}_i \vec{u}_i^T$$ so that $\|A-A_k\|_F$ is minimal. I'd like to do ...
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Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...
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Singular value decomposition: does the choice of eigenvectors matter?

I'm trying to calculate the SVD-decomposition of a certain matrix, i.e. $ A = U \Sigma V^T$. My solution doesn't yield $A$ again; I just can't get the signs correct. I'm wondering if this is just a ...
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SVD parallel for distributed memory

I want to build a code executing a parallel SVD via the one sided Jacobi method within a distributed memory approach (MPI). The only possibility therefore seems to be a block method. Sofar it looks ...
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48 views

How to solve the following quadratic matrix equation?

Solve the following matrix equation in $D$ $$ A=D^{T}(DVD^{T}+\alpha\lambda_{\max}(D^{T}D)I)^{-1}D$$ where $I$ is the identity matrix, $A$ and $V$ are known matrices, $\alpha$ is a known ...
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Maximizing the trace of a complex matrix

Let's say I have the following maximization problem: $max_U{tr(AU)}$ where $A\in\mathbb{C}$ and $UU^\dagger=1$ I know that for $A\in\mathbb{R}$ and $UU^T=1$ the solution is: $U=XZ^T$ where $X$ ...
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Singular values of the product of two (semi)orthogonal matrices

Let's assume we are given two (semi)orthogonal matrices $U_1$ and $U_2$ with dimension $m\times n$ such that $m>n$. The (semi)orthogonality means $$U_1^TU_1 = I_n$$ and $$U_2^TU_2 = I_n$$ but ...
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Proof that left singular vectors in SVD are orthogonal, and proof of low-rank approximation

I've been reading about SVDs and have a couple questions. First, let $A\in \mathbb{R}^{n\times d}$ be a matrix with SVD $U\Sigma V^T$. Let $\sigma_i$ denote the $i$'th singular value, with ...
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21 views

Compute the SVD of $AB$ from the SVDs of $A$ and $B$

Knowing the SVD of $\mathbb{C}^{m*n} \ni A = U_A\Sigma_AV_A$ and $\mathbb{C}^{n*s} \ni B = U_B\Sigma_BV_B$, is there any way to speed up the calculation of the SVD of $AB = U_{AB}\Sigma_{AB}V_{AB}$? ...
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Given A $m\times n$, $\{u_1,…,u_n\}$ ON basis of $R^n$, prove eigenvalues aren't negative [duplicate]

Given $A$ $(m\times n)$, $\{u_1,...,u_n\}$ ON(orthonormal) basis of $R^n$ which are eigenvectors of $A^TA$ with $\lambda_1 , ... , \lambda_n$ eigenvalues accordingly. Prove: Eigenvalues are not ...
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189 views

Minimizing $\|Ax\|_2$ subject to $\|x\|_2 = 1$

I have a Matlab program to estimate a vector $x$ from noisy measurements. I use the singular value decomposition (SVD) to solve the linear equation $Ax=0$ (where the number of equations is greater ...
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39 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
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23 views

invertible subspaces and SVD

Suppose X is a $m\times d$ real matrix. I am trying to prove the following claim: $XX^T$ is invertible iff $ span\{x_1,x_2,...,x_m\}=\mathbb{R}^d $ such that $x_1, x_2, ..., x_m $ are X's columns. ...
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34 views

low rank approximations and diagonalization

I would like to discuss or hear an opinion about the following. Given is the (hermitian) $n\times n$ matrix $A = D+M V M^{\dagger}$ with D diagonal. I would like to calculate the eigenvalues (and ...
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79 views

Prove that $A^2=A\iff \Sigma K=I_r$

Let $A$ be a square complex matrix and let $A=U\Sigma V^*$ be a singular value decomposition. Then $A$ can be written as $$A=U\begin{bmatrix} \Sigma K & \Sigma L\\ 0 & 0 \end{...
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Efficient distributed algorithm for Singular Value decomposition

Is there any distributed algorithm possible for calculating Singular value decomposition of a matrix? As far as if have explored at many places (like here pg-132) it is written that because of high ...
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45 views

why bother with extra orthonormal vector in Singular value decomposition

when we do the SVD for a $m\times n$ matrix, we have to extend the set $u_1, ... , u_r$, to an orthonormal basis $u_1, ... , u_m$ for $R^m$ if $r<m$. But why don't we just fill zero vectors to make ...
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SVD as a solution to linear least squares

I'm a little confused about the various explanations for using Singular Value Decomposition (SVD) to solve the Linear Least Squares (LLS) problem. I understand that LLS attempts fit $Ax=b$ by ...
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37 views

Orthogonalise a matrix with respect to another matrix

I need to do a particular operation and I am not able to find a name or some ideas to make the operation happen. I have a matrix $F_s \in \mathbb{R}^{n \times n_s}$. I do an Singular value ...
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53 views

SVD and Low-rank approximation

In the proof of Low-rank approximation by Trefethen & Bau, It is written: Theorem 5.8 : A is an $m \times n$ Matrix. For every $v$ with $0 \leqslant v \leqslant r$, define $$ A_{v}=\...
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Independent components with a known relationship between sources

I'm performing an experiment, and I expect that my results of the form: $$\mathbf{M}=\mathbf{ce}, \mathbf{M}\in\Bbb{R^{m\times n}},\mathbf{c}\in\Bbb{R^{m\times 2}},\mathbf{e}\in\Bbb{R^{2\times n}}$$...
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106 views

Calculating SVD By Hand

When calculating the SVD of the matrix $$A = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix}$$ I followed these steps $$A A^{T} = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix} ...
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21 views

Clarification between the left and right singular vectors of the singular value decomposition (SVD).

I am hoping for clarification on the difference between the left and right singular vectors of the SVD for some matrix: $A = U\sum V^T$. I'd like to explain how I understand these vectors using the ...
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33 views

Question on SVD Uniqueness Proof

I have problem understanding the proof of uniqueness for SVD by by Trefethen & Bau. If the lengths of the semi-axes of the hyper-ellipse are distinct, then semi-axes themselves are ...
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45 views

Optimal ordering in Jacobi SVD algorithm

In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...
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25 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $...
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SVD matrix properties after blockwise scalar multiplication

Are there any findings relating the singular value decomposition (SVD) of a block matrix to the SVD of the same matrix after multiplying each block with a constant ? In other words, let $A \in \...
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50 views

SVD decomposition

I'm currently looking for the SVD decomposition of $$\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 1 \end{bmatrix}$$ but I am having a struggle, because $A^TA$ has a ...
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Extending $\{u_1, u_2\}$ to an orthonormal basis when finding an SVD

I've been working through my linear algebra textbook, and when finding an SVD there's just one thing I don't understand. For example, finding an SVD for the 3x2 matrix A. I will skip the steps of ...
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25 views

Element with maximum magnitude in $A \leq \max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A

Let $A$ be a matrix with real values. Is it true that element with maximum magnitude in $A$ is less than $\max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A? That is, is $$ \max_{ij} |A_{ij}...
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constrain general solution of ill-posed linear system to $\Re_+$?

I have a solution space of an under-determined linear system Ax = b with n x m matrix A: $$x= x0 + V2 * c (1)$$ with [U, S, V] = svd(A); V2 = V(:,r+1:end); $x0 = A^+ b; $ r = rank(A); I ...
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square eigenvectors for Singular Value Decomposition?

This is from my textbook What I don't understand is, $V$ and $U$ are already square, why the textbook says "if we want to make them square"?
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Rank of product of SVD vectors

Given a matrix $M$, we can compute its singular value decomposition $M=U\Sigma V^*$ where $^*$ is the complex conjugate transpose. $U$ and $V$ are unitary, so $UU^*=I$, $VV^*=I$. Let's take the $i$-th ...
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Perturbations on SVD left singular matrix

Given a symmetric matrix $A\in \mathbb{R}^{n\times n}$, with all the entries greater than zero $A_{i,j}>0$ with rank $k<n$, we can calculate its SVD decomposition: $$ A = USU' $$ Assuming now ...
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Jacobi SVD algorithm implementation

Is this implementation of Jacobi SVD algorithm according to the standard algorithm? Please verify. Is this Hestenes Jacobi method? I have seen pseudo code of Jacobi algorithm like here which ...
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null space from SVD

It is said that a matrix's null space can be derived from QR or SVD. I tried an example: $$A= \begin{bmatrix} 1&3\\ 1&2\\ 1&-1\\ 2&1\\ \end{bmatrix} $$ I'm convinced that QR (more ...
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Possible quick solution of SVD of covariance matrix of Xv, where v may change, while X does not.

I am current trying to work on one algorithm, that for Iteration $t$, I need to calculate the SVD of $(X\text{diag}(v^t))^T(X\text{diag}(v^t))$. This could be very slow if $X$ is of high dimension. ...
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SVD and homogeneous equation

Suppose a $m \times n$ matrix $A$, and column vector $h$. ($A$'s rank is equal or smaller then $n$(=$h$'s length).) If, $$ Ah=0 $$ then $h$ can be the last column of $V$ where $A = UDV^T $. ($UDV^T$...
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Should the correlation PCA projection be computed on original or normalized samples?

Suppose we compute the correlation PCA of a dataset $X$ (with $m$ variables and $n$ observations) by first normalizing the input variables. That is: mean $\rightarrow 0$ and standard deviation $\...
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Nearest singular matrix

Let the SVD of $A \in \mathbb R^{{n}*{n}} $ be given as $A=\sum_{i=0}^n \sigma_{i}u_{i}v_{i}^{T}$ where $\sigma_{1}\gt \sigma_{2}>{...}>\sigma_{n-1}=\sigma_{n}>0 $ Compute a matrix $B$ such ...
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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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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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44 views

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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66 views

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 this ...
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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. ...