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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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Proof of the Single Value Decomposition

I am working through the a proof of the single-value decomposition, from Strang's 'Introduction to Linear Algebra, 4th edition'. I have included the proof as shown in the book at the end of this post. ...
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Is here a mistake or can you explain me what orthonormal basis mean?

I am currently trying to get insight into SVD, and I found one book with an explanation of how we find the 𝑉 and 𝑈 matrices and why it holds that any 𝑚×𝑛 matrix can be represented in this ...
comediann's user avatar
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Best rank r approximation of a matrix with wildcard elements?

Let's say you have a $n \times n$ matrix, for example, this $4 \times 4$ matrix: $$A = \begin{bmatrix}a & b & c & d \\\ e & f & g & h \\\ i & j & k & l \\\ m & ...
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Minimum singular value vs condition number to determine closeness to singularity

Which criteria should I use to determine if a matrix is close to singularity or not? My application is to try to find an optimal matrix that maximizes the minimum singular value/minimizes the ...
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Are all real uniformly bounded matrices orthogonal?

Inspired by a closed question, let $A$ be an integer coefficient matrix such that $\exists B$ and for $i \in \mathbb{N\setminus\{0\}}, \ |a| \leq B$ for all elements of $A^i$. This condition is ...
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my question is about the singular value decomposition [closed]

For a matrix like A after getting its singular-value-decomposition ,How is this mathematical formula σ1 ≥ σ2 ≥ . . . ≥ σn ≥ 0 obtained? In another words, How is it discovered?
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Projection matrices and generalized inverses

I am trying to characterise the projection matrices using the singular value decomposition (SVD) in order to better understand oblique projections. Orthogonal projection A projection matrix $P\in\...
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Optimal rotation matrix

Context: I am trying to adapt the Rigid point set registration algorithm from Point Set Registration: Coherent Point Drift to include rotation information. My problem can be stated as follows: $\max_{...
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Matrix multiplications with SVD

I'm trying to understand the calculation of $SU = U(\sigma^2 I + D^2)$, which I need to prove with the condition $S(\sigma^2 I + WW^\top)^{-1}W = W$. Let $W \in \mathbb{R}^{d \times m}$ and $W = UDV^\...
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Why do we normalize means to 0?

I was wondering why normalization (e.g. for PCA) generally brings the means to 0. I understand the idea behind normalizing --- every variable would contribute the same, regardless of their differences ...
pseudobulbose's user avatar
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Interpretation of QR "values" (a la singular values)?

Let $A\in\mathbb{R}^{m\times n}$ with $m>n$ and $\text{rank}A=n$. There exists $\hat{Q}\in\mathbb{R}^{m\times n}$ with orthonormal columns and $\hat{R}\in\mathbb{R}^{n\times n}$ upper triangular ...
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$3\times3$ real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing $3\times3$ real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate ...
Venom's user avatar
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Distance between subspaces with spectral norm

I was trying to prove this following theorem , Let $$ W=\begin{bmatrix} \underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2} \end{bmatrix} $$ $...
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Singular value decomposition algorithm recommendations for smaller dense matrices

I am looking for recommendations for SVD algorithms for dense matrices. My supervisor specifically requested that I do not use external libraries for this (otherwise I'd likely use LAPACK and call it ...
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Moore-Penrose Pseudoinverse of Augmented Linear Systems of Equations

The problem I am working on is comprised of $N$ independent system of equations with the same size $$ A_{4096\times3}x_{3\times1}=b_{4096\times1} $$ where $x$ has to be found using Moore-Penrose ...
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what happens to singular values with one-rank update?

I have a square matrix $A$ and two vectors $a$ and $b$ such that $\sum a_i = 1$ and $b_j = 1$ for all $j$. I would like to know if there is some way of expressing the relationship between the singular ...
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Extend an orthogonal set of vectors to an orthonormal basis in SVD.

I'm learning about Singular Value Decomposition (SVD) and how to compute each matrix in the decomposition $A=U\Sigma V^T$. I know how to compute $\Sigma$ and $V$ and hence most of $U$, since we know ...
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Is polar decomposition commutative for diagonal matrices?

I did a error while understanding about the polar decompositon. I thought polar decomposition is PU, but it is UP. While trying ...
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How to prove a matrix $M$ has incoherence property?

By incoherence, I am referring to equation 1.18 in the paper The Power of Convex Relaxation: Near-Optimal Matrix Completion Given a rank 1 matrix $n\times n$ matrix $M = x \mathbf{1}^T$, $x \in \...
Nebiyou Yismaw's user avatar
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Proof of first step in SVD

I have to proof the following statement: Prove that for a given $A\in \mathcal M _{n \times m}(\mathbb R)$ there exist two orthogonal matrices $U \in \mathcal O(n)$, $V \in \mathcal O(m)$ such that: $...
Mikel Solaguren's user avatar
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A supremum on orthogonal matrices

I'm working on a problem where I want to find the supremum over the orthogonal group $O_n(\mathbb{R})$ of the sum of the upper triangular elements of matrices in this group, specifically we want to ...
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How does the SVD of a matrix relate to the SVD of multiple stacked duplicates of the matrix?

Suppose I have a matrix $A \in \mathbb{R}^{m \times n}$ with singular value decomposition (SVD) $A = U S V^T$. I then stack $A$ on top of itself $k$ times: $A' = [A; A; ...; A] \in \mathbb{R}^{km \...
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Relation between zero singular values and eigenvalues

Given a non-normal diagonalizable square matrix $A\in \mathbb{R}^{N\times N}$, we know its Eigenvalues $\lambda_i$ and its singular values $\sigma_i, i=1,\dots, N$. Say, I approximate $A$ by $\sigma_i\...
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Are there any numerically stable methods for computing the condition number of a near-singular matrix that are fast?

I know of a number of techniques for computing the condition number that are either slow (doing a whole singular value decomposition) or fast (ratio of norm of matrix and its inverse obtained via LU ...
meisel's user avatar
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Can principal components changed by a normalization method be used to construct original data shape with SVD

I'm planning to use an algorithm called Harmony, designed for data normalization, particularly in the context of single cell data analysis. Harmony operates by taking principal components (PCs) as ...
MadmanLee's user avatar
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Variation of Generalized singular value decomposition with more than two matrices?

I am familiar with the Generalized singular value decomposition (GSVD). In GSVD, given matrices $A_1 \in \mathbb{C}^{m_1 \times n}$ and $A_2 \in \mathbb{C}^{m_2 \times n}$ it is possible to decompose ...
Dawson Beatty's user avatar
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If a matrix is an outer product of two vectors; can I determine the vectors? [closed]

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 ...
MengXing Chen's user avatar
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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 ...
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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_{...
Mark's user avatar
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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 ...
ivan's user avatar
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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 ...
Momo's user avatar
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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 ...
SVDieseas's user avatar
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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....
Heribert Greinix's user avatar
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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 ...
William Deng's user avatar
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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 ...
gwn's user avatar
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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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