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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If a matrix is an outer product of two vectors; can I determine the vectors?

I am working with floating point numbers. There is a 3x3 matrix that has determinant 1e-14. I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how ...
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In sparse ridge regression, why we have this property

In ridge regression, we can estimate $\hat y$=$X(X^TX+\lambda I)^{-1}y$,where $X$ is covariate matrix with n rows and p column. And my teacher says that we can use SVD to rewrite this formula as:$\hat ...
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SVD and least square solution

Let $K \in \mathbb{R}^{m,n}$, $u \in \mathbb{R}^n$, and $f \in \mathbb{R}^m$. Assume that $m < n$ and $K$ have full rank so a solution exists but is not unique. I want to understand why this ...
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Decomposing a matrix with unit sphere constraints [closed]

I would like to decompose an $m\times n$ matrix $A$ into two matrices $U\in\mathbb{R}^{m\times n}$ and $V\in\mathbb{R}^{n\times n}$ such that $UV=A$, and the $m$ rows of $U$ each have unit magnitude. ...
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Reasons of computing smallest eigenvalue $R^TR$ instead of singular value

I have problems in understanding why author of this article uses smallest eigenvalue of a cross product matrix instead of a data matrix. I know that $SVD(AA^T)=UD^2U^T$, but I don't know why not ...
Wilk's user avatar
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Does PCA always find the best-fitting plane?

Here, the best-fitting plane is the plane that minimizes the sum of squared perpendicular distance from the data points to the plane. In other words, the best-fitting plane is the solution to the ...
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Rotation, reflection and scale using SVD for complex matrices

I get that in the case of a real $n\times n$ matrix, $M$, in the SVD of $M = USV^T$, $U$ represents rotation and/or reflection in the input basis, $S$, the set of singular values denotes scaling and $...
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Is there an "orthogonal factorization" of bivariate functions that is analogous to the SVD of matrices?

For a matrix $X \in \mathbb{R}^{m\times n}$, we have the SVD decomposition $$ X = U D V^\top, $$ where $U\in\mathbb{R}^{m\times r},\ V\in\mathbb{R}^{n\times r}$ are orthonormal matrices and $D=\text{...
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Test for rotational component in arbitrary matrix

I am studying differential forms and I am trying to characterize exterior derivatives. This journey keeps taking me back to linear algebra and my most recent insight has been the Singular Value ...
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SVD decomposition of a square matrix of complex numbers

Le $M$ be any matrix in $C^{n \times n}$. Consider the matrix $MM^*$. This matrix is Hermitian ($(MM^*)^* = MM^*$), and positive semi-definite ($\forall v^*, v^*MM^* v = (v^* M) (M^* v) = (v^*M) (v^*M)...
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How to find a matrix perturbation which lowers the rank of a matrix

I have a matrix $A \in \mathbb{R}^{m x n}$ which has independent columns. I want to find the smallest perturbation which will make it have a kernel and a vector in that kernel. Something like $$ \min_{...
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Properties of the eigenvalues of $A \cdot B$

Take two (real) matrices $A$ and $B$, where both have (real) eigenvalues within the unit circle, $B$ is also a diagonal matrix. Can I say something about the eigenvalues of the product $A \cdot B$ ? I ...
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How do I normalize data before SVD?

Consider a satellite orbiting earth taking images. Since the problem can be approximated by the satellite moving with constant velocity and orientation relative to the fixed ground; the projection ...
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Decomposition of a matrix into observability and controllability matrices

$\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}$ I have a matrix $\boldsymbol{Q} \in \mathbb{R}^{M \times M}$ in ...
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Singular value decomposation orthogonal to another matrix

I have two real matrices $A^{k\times m}$ and $B^{k\times n}$, let's assume $k\gg m$ and $m>n$. Let's also introduce an augmented matrix $C = [A \quad qB]$. I want to get the 'almost' singular ...
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Discrepancies in Custom SVD Implementation Compared to np.linalg.svd - Sign Issues

I've been working on implementing a Singular Value Decomposition (SVD) algorithm from scratch in Python without using the np.linalg.svd function. My goal is to understand the underlying mechanics of ...
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Solving A = RB using SVD

I have a linear system given by $\mathbf{A} = \mathbf{R}\mathbf{B}$ where $\mathbf{A}$ and $\mathbf{B}$ are 3-by-n matrices and $\mathbf{R}$ is a 3-by-3 orthonormal matrix (i.e., a rotation matrix). ...
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Proof that if eigenvalues are equal to singular values then it is symmetric positive-semi definite

Take $A\in R^{kxk}$. Suppose that its eigenvalues are equal to its singular values. Then show that $A$ is symmetric and positive semi-definite. I've found sources stating it but I haven't managed to ...
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Deriving the SVD from the eigendecomposition

If $A$ is a rectangular matrix of dimensions $m\times n$, then $S_L=AA^T$ and $S_R=A^TA$ are square symmetric matrices. Hence, using the eigendecompostion we can write $$ S_L=AA^T=U\Lambda_{S_L} U^T $$...
ady's user avatar
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What to do when the Gram matrix of an underdetermined system is singular?

I am currently trying to speedrun my linear analysis course, I have been doing pretty well so far, but hit a wall when the lectures started hitting on SVD and under/overdetermined systems of equations....
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Geometric interpretation of left-singular and right-singular vectors

I wanted to ask if and how $A^{T}A$ respectively $AA^{T}$ can be interpreted geometrically in the sense of it's eigenvectors being the left and right singular vectors? What is the geometric ...
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Most General Solution of a Matrix Equation (Arising From SVD)

Suppose we have an arbitrary but known $n\times m$ complex matrix $A\in\textbf C^{n\times m}$ which therefore has an $m\times n$ conjugate transpose $A^{\dagger}\in\mathbf C^{m\times n}$. Now suppose ...
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Confusion regarding the geometrical meaning of singular values in SVD

I am trying to visualize in MATLAB the relationship between the singular value decomposition (SVD) of a matrix of points. To simplify the problem, I am working in 2D and I am considering an ellipse ...
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Efficient SVD of low-rank matrix of the form $C=AB^{T}$

Let $A,B$ be two real matrices, of dimensions $n \times k$ and $m \times k$, respectively. I assume that $k \ll n,m$. I am interested in computing the SVD of the product matrix $C = AB^{T}$. The ...
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Why does scaling a column of $U$ by $-1$ make the SVD of a matrix incorrect?

Consider the SVD of a matrix $A = U\Sigma V^T$ where $U = [u_1 \; u_2 \; u_3 \; ... \; u_m]$. If I scale a column of $U$ by $-1$ to, for example, $U^{\prime} = [-u_1 \; u_2 \; u_3 \; ... \; u_m]$, $U^{...
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Why the singular values of $A$ is the positive square root of the eigenvalues of $A^{\top}A$?

I've tried to look for the answer to this curiosity over the internet, but I couldn't find any explanation that is clear enough. So here's the question: consider a linear transformation $L\colon\...
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Equality singular value $\sigma_1(A)=\sup_{\lVert x\rVert=1}\lVert Ax\rVert=\sup_{\lVert x\rVert =1,\lVert y\rVert=1}\langle Ax,y\rangle$

I am trying to solve: $\sigma_1(A) = \sup\limits_{\lVert x \rVert = 1} \ \lVert A x \rVert = \sup\limits_{\lVert x \rVert = 1, \lVert y \rVert = 1} \langle Ax,y\rangle$ where $\sigma_1(A)$ is the ...
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Smoothly choosing SVD

Let $X$ be a connected open set of $\mathbb{R}^2$ and $f:X\rightarrow GL_2(\mathbb{C})$ be smooth. Under what conditions can we smoothly choose a singular value decomposition, i.e. find smooth maps $U,...
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Singular Value Decompositon

In singular value decomposition, where A = U$\Sigma$V and A is some n x m matrix, n >= m and rank(A) = m. can we conclude that the pseudoinverse of $\Sigma$ multiplied by $\Sigma$ is equal to the ...
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Phase degeneracy of singular vectors in SVD of complex matrix

An $m*n$ complex matrix $M$ is given. I take its SVD, $M=U\Sigma V^\dagger$. My question is, are the singular vectors $\textbf{u}_n$ and $\textbf{v}_n$ degenerate? Can I pick any phase for one that ...
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When is the largest singular value is strictly greater than all eigenvalues?

I found a proof of the fact that the largest singular value is always greater or equal to $|\lambda|_{\max}$. In what cases equality holds and in what cases it is strictly greater? Thank you.
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Weighted Nearest Kronecker Product

For a given $m n$-length vector $a$, the problem of finding an $m$-length vector $x$ and an $n$-length vector $y$ that minimize $$ \lVert a - x \otimes y \rVert^2 $$ is known as the Nearest Kronecker ...
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Computation of $\phi\phi^T$ where $\phi$ is a vector that depends on semi-orthogonal matrices.

Let $r \leq \min(m,n)$ and $U \in \mathbb{R}^{m \times r}$, $V \in \mathbb{R}^{n \times r}$ be matrices such that $U^TU=V^TV=I_r$. Let also $U_{\perp} \in \mathbb{R}^{m \times (m-r)}$ and $V_{\perp} \...
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Scale Invariant Singular Value Decomposition

I am looking for a reference to the concept of Scale Invariant SVD which is mentioned in the "variations and generalisations" section of the Wikipedia article for SVD: https://en.wikipedia....
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Linear Least Squares using SVD for rank deficient case

For the linear least squares problem: $\min_\boldsymbol{x}{\|\boldsymbol{y}-A\boldsymbol{x}\|_2^2}.$ Using the SVD of $A\in\mathbb{R}^{m\times n}(m\geq n)$ given as: $A=\sum_{i=1}^r\sigma_i\boldsymbol{...
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Is this perturbation of a rank $r$ matrix still of rank $r$?

Let $M \in \mathbb{R}^{m \times n}$ be a matrix of rank $r$ with compact SVD $M=U\Sigma V^T$ ($U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$ are semi-unitary matrices and $\Sigma$ ...
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Upper bounding the infinity norm of a quantity involving semi-unitary matrices

Let $M \in \mathbb{R}^{m \times n}$ be a matrix of rank $r$ with compact SVD $M=U\Sigma V^T$ ($U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$ are semi-unitary matrices and $\Sigma$ ...
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Proof of alternative writing of SVD

I'm trying to understand Singular Value Decomposition. So far I have seen the regular notation for the SVD of a matrix $A$ as \begin{equation} A=USV^T \end{equation} But there is a second ...
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Why is the expectation of the inverse of product of random matrices with entries i.i.d. Gaussian the identity matrix multiplied by a constant?

I'm reading Stanford's EE270 lecture notes on Large Scale Matrix Computation and Optimization found at: https://web.stanford.edu/class/ee270/scribes/lecture8.pdf In Section 8.7, there is the ...
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SVD of "normalized" SPD matrix: $P=\sigma \rho \sigma^T$. Relate SVD of $rho$ to the SVD of $P$?

$P\in\mathbb{R}^{n \times n}$ is a symmetric positive definite (SPD) covariance matrix, and can be factorized into standard deviations and correlations as $P=\sigma \rho \sigma^T=\sigma \rho \sigma$. ...
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SVD or Cholesky on sum of SPD matrices

Let $A$ and $B$ be symmetric positive definite (SPD) matrices and $C=A+B$. I know the SVD or Cholesky decomposition of A and B, $A=U_A\Sigma_AU_A^T=L_AL_A^T$ and $B=U_B\Sigma_BU_B^T=L_BL_B^T$. Can I ...
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SVD of complex matrix and real-valued representation

Consider a matrix $A \in \mathbb{C}^{m\times n}$. The SVD of $A$ reads: \begin{equation} A = U\Sigma V^H \end{equation} where $U \in \mathbb{C}^{m\times m},V \in\mathbb{C}^{n\times n}, \Sigma \in\...
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Getting largest eigenvalue of A'WA for diagonal W

A is a large ill-conditioned matrix that is available only as a function performing matrix-vector products, and a diagonal (weight) matrix W that is full rank. The eigenvalues of A'A are widely spread ...
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Is there a general formula for SVD of inverse matrix?

Suppose $A$ is a square invertible matrix and let $A=USV^{*}$ be a SVD of $A$. (Note that $U,V$ are unitary and $S$ is diagonal with descending entries on diagonal) At first, I thought SVD of $A^{-1} =...
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3D fitting with SVD - Uncertainty estimate

I am trying to use SVD to fit a particle track through a detector and getting the direction vector from the right singular matrix. The matrix $A$ is defined by the positions $\vec{p}_i$ of the signals ...
krixikraxi's user avatar
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Simple method for calculating a cylinder that best fit to points (least squares, SVD, or whatever)

Say I have an infinite cylinder that can be defined by its axis $\pmb v$, radius $r$ and a center point $\pmb c$ indicating its position. Now, say I have many points in 3D and I want to find the model ...
ebernardes's user avatar
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Proving that $\Vert A-A_k\Vert =\sigma_{k+1}$ for SVD

Im trying to prove that $\Vert A-A_k\Vert _2 =\sigma_{k+1}$ for SVD with the usual notation that sigma represents singular values and if $A=\sum_{i=1}^r \sigma_iu_iv_i^T$, then $A_k=\sum_{i=1}^k \...
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How to get a solution in overdetermined matrix equation with zero-vector?

I have searching lots of pages but cannot fine clear answer. I want to solve the matrix equation " $Ax = b$ " with in $A$ is a 10 x 6 matrix with all-known values $x$ is a vector(6x1) which ...
Math_ter's user avatar
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Variational representation of matrix operator norm

Let $W \in \mathbb R^{n\times d}$ and $\mathbb M = \{\Theta \in \mathbb R^{n\times d} | \text{rank}(\Theta)=1, \|\Theta\|_F = 1\}$ where $\|\cdot\|_F$ is the frobenius norm. Show that the operator ...
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Weird hexagons distribution of singular vectors of random matrices.

I ran the following numerical experiment: Sample 10⁶ random Gaussian 2×2 matrices ($A_{ij}∼𝓝(0, 1)$) Compute SVD $A = UΣV^⊤$. Let $u_\max, v_\max$ be the singular vectors of the maximum singular ...
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