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 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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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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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} ...
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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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Condition number plane fit

I fit a 3D plane to 3D points. I setup the corresponding linear system $Ax=0$ by removing the mean of all the points and stacking them as rows into $A$, and solve for a non-trivial ($x\neq0$) using ...
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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 $. ...
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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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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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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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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 ...