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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Is there a name for the matrices with the same eigenvectors?

I am given a symmetric positive semi-definite matrix $A\in\mathbb{R}^{n\times n}$. I perform the SVD, $$A = V \Sigma V^\top,$$ where $V$ is unitary and $\Sigma$ is diagonal. Next, I pick up another ...
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Determining A Vector Through the Center of Multiple Points on a Sphere

I am working on a machine vision task that requires me to determine spin rate and spin axis of a moving ball. I have had some luck, and actually do have a solution but am looking for a more efficient ...
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Why are Latent Factor Models called "SVD"?

Latent factor models (usually used for recommender systems) are a matrix decomposition of matrix $R$ such that $$ R = P \cdot I^T $$ with the "twist" that values in $R$ can be missing (which ...
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Finding the best approximation of orthonormal vectors has the largest square inner product with a set of given vectors

Suppose we are given a set of fixed vectors $v_1,\dots,v_m\in\mathbb{R}^n$ and assuming $m\leq n$. I want to find an orthonormal set $[u_1,\dots,u_m]$ such that the next quantity is maximized $$ \...
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SVD of a Block Matrix

Given a block matrix $A=\left(\begin{array}{cc} \alpha & b^H \\ a & M \\ \end{array} \right)$, where $\alpha$ is a complex number, $a,b$ are two complex vector of dimension $n$, and $M$ ...
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Constructing a special kind of SVD

Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as $$ A = XD_AY^H \\ B = XD_BY^T $$ where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
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Can any pair of SVD matrices diagonalize other matrices which share at least one pair of SVD matrices?

The title might require a little explanation. Given a complex matrix $Q\in\mathbb{C}^{n\times n}$ it has a singular value decomposition $Q=UD_QV^T$ with unitary matrices $U$ and $V$ and real, positive ...
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How to compute principal components for a curvature found given XYZ points?

I have a certain XYZ set of points that make up an object. I chose a random point and make the nearest radius analysis and find the neighbors. From these neighbors, I get the green pointcloud curve ...
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projection of the data along the 1st k principal components

I'm a final year maths undergrad doing a course in multivariate data analysis, but I'm really struggling with the linear algebra. In particular the “projection of the data along the 1st k principal ...
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Singular vectors from true and sample covariance do not align

Suppose we have a covariance matrix $C\in\mathbb{R}^{p\times p}$ which is diagonal with exponential decaying values. This covariance matrix is used to sample $n$ data points $x\sim\mathcal{N}(0,C)$ ...
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Find the best fitting plane given n points that goes through the origin

My Problem I need to find the plane which best fits a given number of points (at least 3) and that must contain the origin. I'm supposed to do this with the least squares method, but i can also use ...
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On the uniqueness of SVD in the $2$-dimensional case

On page 155 of Tristan Needham's Visual Differential Geometry and Forms, the singular value decomposition (SVD) is given by $$M = R_{\phi} \circ \Sigma \circ R_{-\theta}$$ with the associated picture: ...
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Uniqueness of monic polynomial that can achieving $\inf_{p_n \in P_n} ||p_n(A)||>0$ with $A$ being an nonsingular matrix.

Problem: Let $A$ be an $m \times m$ nonsingular matrix. Suppose $\inf_{p_n \in P_n} ||p_n(A)|| = ||p^*(A)|| > 0$ where $P_n$ denotes the set of all degree-n monic polynomials: $$P_n ={p:p\text{is a ...
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Polar decomposition of a linear combination of unitary matrices

Consider a complex-valued square matrix $M$ of the form $$M = \frac{1}{2}\left(U_1 + e^{-i\phi}U_2\right),$$ where $U_1$ and $U_2$ are unitary matrices and $\phi$ is a real number. Moreover, consider ...
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Moore-Penrose pseudoinverse solves the least squares problem (SVD framework) [duplicate]

I am a computer science researcher who has to learn some numerical linear algebra for my work. I have been struggling with the SVD and Moore-Penrose pseudoinverse as of late. I am trying to solve some ...
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With $UD V^T$ is the SVD of $X$, is $X\mapsto UV^T$ Lipschitz over the set of matrices with full rank and smallest singular value at least $a>0$?

Let $m\le n$. For any matrix $X\in R^{m\times n}$ with full-rank $m$, define the usual SVD $X = UD V^T$. Define the open set $M_a = \{X\in R^{m\times n}: \lambda_{\min}(XX^T)> a\}$ where $\lambda_{\...
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Question about introduction of variables in proof of SVD in Trefethen and Bau.

I have understood as far as marked with the red line. I am trying to understand the proof that there exists an SVD for all matrices. I don't understand why there is needed for an orthonormal extension ...
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Deriving SVD from polar decomposition

The Wikipedia article on the polar decomposition states that, for any matrix $A \in \mathbb{R}^{m \times n}$, the polar decomposition is defined as $A = UP$ where $U \in \mathbb{R}^{n \times m}$ and $...
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Rotation Invariance of the Distribution of $\ell_2$-norm square of Gaussian Vector

In a paper I'm reading I saw the argument below: Let $G\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and $A$ is a square matrix with SVD $A=U\Sigma V^T$ where $\Sigma=diag(\sigma_1,\dots,\sigma_n)$. Then $|...
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Geometric interpretation of $A^TA$

For a transformation $A \in \mathbb{R}^{n\times m}$ what exactly is the geometric interpretation of the transformation $A^TA$. If I understand it correctly the entries of $A^TA$ are the inner products ...
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What does singular value decomposition of covariance matrix represent?

I am reading the paper "Understanding dimensional collapse in contrastive self-supervised learning". Authors identified a dimensional collapse phenomenon, i.e. some dimension of embedding ...
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Image Deblurring Resources?

I have been researching about image deblurring methods and would like to know if anyone had some good entry points to this. I am looking for references about basic image deblurring, and mainly the ...
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A matrix decomposition problem similar to the CS decomposition

For two column orthonormal $X,Y\in\mathbb{C}^{n\times k},k<n$, prove that there exist unitary $Q\in\mathbb{C}^{n\times n},U,V\in\mathbb{C}^{k\times k}$, such that $QXU= \begin{bmatrix} I_k\\ 0 \end{...
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Matrix as a dataset vs a transformation

In my machine learning class, we are learning about Singular Value Decomposition (SVD) on a data matrix. SVD allows us to "decompose" a $nxm$ matrix, $A$, into a product of three "...
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Upper bound on normwise relative error

Suppose I have two vectors $y_1, y_2 \in \mathbb{R}^{n\times 1}$ which are both transformations of the same vector $x$ by two different matrices $T_1, T_2 \in \mathbb{R}^{m\times n}$, i.e.: \begin{...
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$ \exists L>0, \forall m,n\in\mathbb{N}, \forall A,B \in \mathbb{R}^{n\times m}, \|\sqrt{AA^T}-\sqrt{BB^T}\| \le L \|A-B\|$?

Is it true that \begin{equation} \exists L>0, \forall m,n\in\mathbb{N}, \forall A,B \in \mathbb{R}^{n\times m}, \|\sqrt{AA^T}-\sqrt{BB^T}\| \le L \|A-B\|, \end{equation} where $\|\cdot\|$ is the ...
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Is there a formula for the norm of an orthogonal projection?

In all introductory linear algebra texts there is a discussion on orthogonal projection. Let $u = w_1 + w_2$, where $w_1$ is the projection of $u$ along $v$ and $w_2$ is projection of orthogonal to $v$...
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Why does this algorithm for the Takagi factorization fail here?

In this paper it was found that the Takagi factorization of a complex symmetric matrix $A$ can be calculated from the singular value decomposition $A=U\Lambda V^H$ by $$ A = X\Lambda X^T \tag{1}$$ ...
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SVD of an orthogonal projector

Here is my observation: Suppose there is an orthogonal projector $P$ such that $P=P^2$. Then for arbitrary $x$, $Px$ and $(I-P)x$ are orthogonal. So we have $$ x^* P^* (I-P)x=0$$ where $A^*$ means ...
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How does the SVD behave under a "phase transformation"?

The singular value decomposition of a complex matrix $A$ can be written as $$ A= UDV^{H} $$ where $U$ and $V$ are hermitian matrices and $D$ is a diagonal matrix with entries $(D)_{ii}=\sigma_i\geq0$ ...
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Some questions regarding svd computations

0 I've some questions regarding svd: Consider a matrix A ∈ R500×5 and its SVD [U, S, V T ] = svd(A). (Assume A is centered). a) is the second left singular vector of A is the direction in R5 with the ...
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Singular value decomposition for a convolution kernel

I am trying to get some insight into how to solve the following linear equations: $$ \frac{dA_n}{dt} = i\sum\limits_{m} B^*_{n-m} C_m$$ $$ \frac{dC_n}{dt} = i\sum\limits_{m} B_{n-m} A_m$$ where $B_n$ ...
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Positive lower bound for minimum magnitude of singular value of a rectangular matrix

Fix $\Gamma_0 \in \mathbb{R}^{k\times d}$ and $C_{\Gamma} \in \mathbb{R}$. If necessary, we can assume $C_{\Gamma} $ is a very small positive number. we've proved that $||\hat{\Gamma}- \Gamma_0||_1 \...
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What does it mean that the singular values is approximately $\mathcal{O}\left(i^{-p-1 / 2}\right)$?

Could you please explain the following statement (last sentence): The "smoother" the kernel $K$, the faster the singular values $\mu_{i}$ decay to zero (where "smoothness" is ...
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Using SVD to write an eigendecomposition

Let $A\in\mathbb{R}^{n\times n}$. Use the SVD of A to write down an explicit eigendecomposition of $$H = \begin{bmatrix}0 &A^{T}\\A & 0\end{bmatrix}.$$ Hint: If $\sigma$ is a singular value of ...
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Prove a part of the power iteration algorithm for finding SVD

I hope you can lead me to the solution or at least how to start because im a little confused here. Let $A\in\mathbb R^{m\times n}$ and define $C=A^tA$ with $\lambda_1\geq..\geq\lambda_n$ eigenvalues ...
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Best optimization technique for solving overdetermined systems with a constraint

I am trying to make a prediction model based on a system of linear equations: $A\vec{x}=\vec{b}$, where $\vec{x}$ ($m\times1$) is my learning parameters, $A (m\times n)$ and $\vec{b}$ $(m\times1)$ are ...
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Singular Values of Projection matrix

In this question, someone proves that singular values of projection matrix must larger than 1: A projection $P$ is orthogonal if and only if its spectral norm is 1 Is it a right rule of singular ...
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Condition number of block matrix

Given a block matrix: $$A= \begin{bmatrix} D & E^T \\ E & 0 \end{bmatrix} , $$ where $D$ is a diagonal, positive entry matrix and $E$ is an incidence matrix of a graph, how is the condition ...
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How to prove SVD can give us the best rank-$k$ approximation under unitary invariant norm?

Let $\|\cdot\|$ be a unitary invariant norm on $\mathbb{R}^{m \times n}$. $\forall A \in \mathbb{R}^{m \times n}$, suppose the SVD of $A$ is $$A = \sum_{i=1}^{r} \sigma_i u_i v_i^{\mathrm{T}}$$ where $...
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Singular Value Decomposition (SVD) with Monotonic Constraint

I am trying to compress some cumulative distribution functions (CDFs) which are stored in an $N \times M$ matrix $A$. Each of the $M$ columns contains $N$ monotonically-increasing values which might ...
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What does "up to complex signs" mean?

I'm studying the SVD, but I'm confused about the term "up to complex signs" If A is square and the singular values are distinct, the left and right singular vectors are uniquely determined ...
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Singular value decomposition for linear transformations (operators)

I find that the singular value decomposition expression is often written for a matrix instead of a linear transformation. Accordingly I have the following (fairly basic) question on my mind. If $A$ ...
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Matrix inverse step in SVD & ridge regression

When we do OLS of $y$ on $X$, with $X$ being a n x p input matrix, the OLS $\beta$ is $(X^TX)^{-1}X^TY$, and the Ridge regression beta is $(X^TX+\lambda I)^{-1}X^TY$. Also, the singular value ...
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SVD and eigenvectors

Let $A = U \Sigma V^{H}$ be the SVD. I must show that columns $v^{1}, \dots, v^{n} \in V$ form a complete set of eigenvectors of $AA^{H}$. Note that $\text{rank}(A) = r$ and $m \ge n \ge r$. I simply ...
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Estimating the right singular vector corresponding to an all positive left singular vector.

Suppose that $A$ is a $k\times n$ real matrix (and for intuition suppose that $k$ is very large) that has SVD $U\Sigma V^\dagger$. Suppose further that we know that there is exactly one column of $U$ ...
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SVD of residual matrix non-orthogonal to orthogonal projection?

Suppose we have two data matrices, $X$ which is a $m$ x $n$ matrix, where $m$ >> $n$ $Y$ which is a $n$ x $p$ matrix, consisting of $p$ orthonormal columns, where $n$ > $p$ Next, we find ...
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Why should unit vectors that get linearly mapped to the semi-axes of an ellipse be orthogonal?

I'm reading Wikipedia's visual proof of the singular value decomposition. They say: To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces –...
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Low-rank approximation of Hermitian matrix with given spectral decomposition

Given a Hermitian matrix $C^{n \times n}$ with a known spectral decomposition $U \Delta U^{-1}$. Is there any way to do a low-rank approximation of $H$ without computing the SVD of $H$ from scratch?
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Derivative of a matrix function that applies on the singular values

Let $F(A)$ be a matrix-valued function, operating on real-valued matrix $A \in \mathbb{R}^{m, n}$ that applies a scalar function $f(\lambda)$ on the singular values of $A$. That is, suppose $A$ has ...
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