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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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Model reduction of estimated state space models - System identification

Assume that we have a dynamical model in form of this simple transfer function $$G(s) = \frac{1}{2s^2 + 5s + 4}$$ G = tf(1, [2 5 4]) We do a step response with ...
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Rank of covariance matrix

I am having a problem with rank deficiency in a covariance matrix. I have a data-set of M variables and N observations, M>N. Calculating the singular value decomposition of the data-sets covariance ...
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Gradient of Rotation matrix estimated via SVD, wrt parameters

I have a matrix $M$ computed as follows: $$ M = PDQ $$ $D$ is a diagonal matrix. $M$ is not necessarily symmetric. I estimate a rotation matrix from $M$ as follows: $$ \left(U,S,V^T\right) = \mathrm{...
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SVD and PCA - help

I have some data on cars: i have been given the Noise,size,speed,if its electric or not, if its a lorry or not. it looks something like this: $$ \begin{matrix} Noise & Size & Speed &...
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Understanding singular value decomposition example

I wanted to view SVD in action (using Octave) by running it on an image and then breaking it down into a set of rank 1 matrices. I'm getting stuck before that though, because I'm unable to reproduce ...
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Norm of a matrix and its rank-1 approximation

Problem This looks like a stupid question, but the following is not straightforward to me $$ \Vert \mathbf{W}\Vert_p \geq \Vert \mathbf{\tilde{W}}\Vert_p, \Vert \mathbf{W}\Vert_F \geq \Vert \mathbf{\...
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Dimensions of $U$ in SVD

In SVD we have $$M=U\Sigma V^T$$ where the columns of $U$ are the eigenvectors of $MM^T$. If $M$ is $m \times n$, is it necessary that $U$ be $m \times m$ or can it be $m \times r$? In other words,...
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Creating a correlated distribution with an indefinite matrix

For the past week, I've been trying to figure out how to implement a linear transformation of a random variable vector with entries $h_{i}$ of the form $\tilde h = Wh$ where h is a non correlated ...
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Every real matrix $A$ is the linear combination of $4$ orthogonal matrices

Question: Prove that every matrix $A\in M_n(\mathbb R)$ is the linear combination of $4$ orthogonal matrices $X, Y, Z, W$ , i.e. $A=aX+bY+cZ+dW$ for some $a,b,c,d\in\mathbb R$. This problem is ...
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Computation of $k$ dominant right singular vectors without SVD computation

I have a maxtrix ${\bf A} \in \mathcal{C}^{m \times n}$, where $m < n$. However, the $m$ and $n$ are large numbers (for eg: m = 50, n = 250). I need to find the $k$ dominant right singular vectors ...
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Number of solutions of $X^\top X = B^\top B$

Let $B \in \mathbb{R}^{d \times n}$ with $d \geq n$ and $B^\top B$ is not singular. How many solutions does the equation $$ X^\top X = B^\top B $$ have? If we do SVDs then $X = U_X D_X V_X^\top$ and $...
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Minimum singular value of sum of rotations

Consider orthogonal matrices $Q_1, Q_2 \in \mathbb{R}^{d \times d}$, where both matrices are proper rotations of angle $\theta_1, \theta_2$ around different axes. Now, consider the symmetric matrix $$...
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What's the reason to use Singular Value Decomposition instead io $(A^TA)^{-1}A^T$ for pseudo inverse?

I wonder what's the reason to use this formula from Singular Value Decomposition $$ A = U\Sigma V $$ $$ A^{\dagger} = V\Sigma^{-1}U^T $$ Instead of $$ A^{\dagger} = (A^TA)^{-1}A^T $$ Both give ...
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Singular Value Decomposition of a rank 1 matrix

I understand that when I do SVD of a rank 1 matrix constructed as $xx^T=U\Sigma U^T$, where U=$\frac{x}{\Vert{x\Vert}}$. But when I calculate $UU^T$ I do not get the identity matrix. What am I doing ...
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Finding SVD of a linear operator (in matrix form)

The linear operator $T\in \mathcal{\mathbb{R}^2}$ defined by $T(x,y)=(2y,x)$ has singular value decomposition (SVD) $$T(x,y) = 2\langle (x,y), (0,1)\rangle (1,0)+1\langle (x,y),(1,0)\rangle (0,1).$$ ...
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Machine learning: What method should I use for classification?

I post this on Math Stack Exchange instead of Data Science Stack Exchange because I want to have the theory, not Pyton import. Assume that we have a vector who contains decimal values, sorted. $$V =...
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Is there any way to use matrix decomposition for finding $A^n$?

If I want to take the power of matrix $A$ with e.g 3, $A^3$ or with power of $-\frac {1}{2}$, e.g $A^{-\frac {1}{2}}$ etc. Is there an easy way to solve $A^n$, where $n\in R$ and $A \in R^{nxn}$ by ...
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If $A = BQ^T$ where $Q$ is an orthogonal matrix, then is $\text{Col}(A) = \text{Col}(B)$?

I have to find a reduced SVD, A = UrDVrT, for the matrix A = BQT. They give me the matrix B and the null-space of $A$. I've already found the singular values for A and the matrix Vr (with an ...
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How far from a matrix the rank does not decrease?

Let $A$ be a real $n \times n$ matrix, and suppose that $\text{rank}(A)=k$. Consider $$ r_0=\sup \{r>0 \,\, | \,\, |B-A| \le r \Rightarrow \text{rank}(B) \ge k \}.$$ Does $r_0$ depend only on ...
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Approximation of a positive definite matrix

I have a covariance matrix (A), which is positive definite. I would like to approximate matrix A by another positive definite matrix B in such a way, that the eigenvalues of B span only 2 orders of ...
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Can I find eigenvalues for $A$ if I know the eigenvalues from $AA^T$

Let's say we have a real matrix $A$ and I find the eigenvalues for $\sqrt{AA^T}$. Is it possible for me then to find the eigenvalues for $A$ without using $$det(A-\lambda I)=0$$ ?
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What is the easiest way to solve SVD numerically?

I want to solve Singular Valude Decomposition(SVD) $$A = USV^T$$ I have been using the QR-method before, but it takes lot of time and is very slow. Is there any easier method to use? Is it easy to ...
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Eigenvalues and -vectors of the marginal distribution

Can the eigendecomposition of a marginal distribution ($\boldsymbol\Sigma'V'=V'\Lambda'$ with $\boldsymbol\Sigma'$ a submatrix of $\boldsymbol\Sigma$) be easily derived from the eigenvectors and ...
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An inequality for a quadratic form and an inner product, and its relationship to the singular value decomposition

Let $A$ be a $p \times q$ real matrix of full column rank $q$, and let $u, v$ be two real vectors of (euclidean) norm 1. I want to know whether the following inequality holds: $$ v^TA^TAv \geq v^T A^...
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SVD of subspace product

$U$ and $V$ are two $d \times d$ orthogonal matrix, $U U^\top = I_d, V^\top V = I_d$. Denote $U_k$ a $d \times k$ matrix with first $k$ columns of $U$, and $V_k$ the same. Denote $U_{-k}$ a $d \times ...
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Going from singular value decomposition to Schmidt decomposition

I'm trying to understand the proof of the Schmidt decomposition found on the Wikipedia page. In particular, the part that I've circled in red in this screenshot: I've tried writing out $U_1, \Sigma, ...
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What is the accuracy of SVD in 3d transformations

I have a triangle $x$ with points $x_1,x_2,x_3\in\mathbb{R}^3$ that were measured at one location. The triangle was then transformed to $\bar x$ where the points were measured again as $\bar{x}_1,\bar{...
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Additivity of nuclear norm for projections

Let $A,B\in \mathbb R^{m\times K}$ with $B=U\Sigma V^T$. Let $r=\operatorname{rank} B$, $(u_1,\ldots,u_m)$ be the columns of $U$, and $S_1=\operatorname{span}(u_1,\ldots,u_r)$. Similarly let $(v_1,\...
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Using SVD to generate a transformation for calibration

Given the problem of trying to find a transformation matrix from one camera to another on a vehicle in a calibration room with the usual checkerboard floor and walls, we can capture images from both ...
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On an inequality involving operator norm of matrices and singular value

Let $A, E \in M_n(\mathbb C)$ be as in this question On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$ . How to prove that $\dfrac {||A^{-1}b-(A+E)...
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On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$

Let $A,E \in M_n(\mathbb C)$ . Suppose $\sigma_\min >0$ be the smallest singular value of $A$ and $||E||_2 < \sigma_\min$. Suppose $||A^{-1}E||_2 <1$. Then how to show that $A+E$ is ...
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$A \in \mathbb{C}^{m\times n}$,$A=FG^*$ and $r(A)=r(F)=r(G)$. Prove $A^\dagger = G(F^*AG)^{-1}F^*$ and $A^\dagger = (G^\dagger)^*F^\dagger$

Let $A^\dagger$ be a Moore-Penrose inverse of a matrix $A$. If $A \in \mathbb{C}^{m\times n}$ and $A=FG^*$, for some $F,G$ and $r(A)=r(F)=r(G)$, prove that $$A^\dagger = G(F^*AG)^{-1}F^*$$ and $$A^\...
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Full and reduced SVD of a 3x3 matrix.

I currently studying for an exam, and I'm currently working my way through some old exam problems and I'm currently at the following. First, we have a matrix $A= \begin{bmatrix} 2&0&0\\2&...
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Why SVD is not unique but the Moore-Penrose pseudo inverse is unique?

I feel confused about the uniqueness of the Moore-Penrose inverse generated from SVD. For any matrix $A$, if $X$ satisfied $$AXA=A, XAX=X, (AX)^\mathrm{T}=AX, (XA)^\mathrm{T}=XA $$then $X$ is called ...
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Search for projection on a special matrix space with regard to Frobenius norm(computer vision background)

Background Define essential space as $$\varepsilon=\{E \in \mathbb R^{3\times3}|E=\hat{T}R\}$$ $$\hat{T}\in\{S\in \mathbb R^{3\times3}|S^T=-S\}$$ $$R\in\{A\in\mathbb R^{3\times3}|A^TA=I,\det(A)=1\}$$...
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Decomposition of a symmetric matrix $xx^T$ into a rank one and residual matrix?

Suppose we have $M=xx^T$ where $x$ is a random vector in $\mathbb{R}^n$. Also, we know that $x=q+e$ where $q$ is distributed according to $D$, i.e., $q \sim D$ and $e$ is a bounded vector. Therefore, $...
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Singular Values of Symmetric Matrix

I saw the following claim in this thread: How to compute the SVD of a symmetric matrix? Claim: The singular values of a symmetric matrix $A$ are the absolute values of its eigenvalues. I ...
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Relation between SVD and POD

I understand that POD is about choosing an optimal base, and i have found this Eckart Young theorem And also have encountered on a book that given an matrix $A$ its projection on POD modes is given ...
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Frobenius norm of $||AA^+ - I||_F = ? $

I need to find a value for the following norm $||AA^+ - I||_F$, where: $A^+$ is the Moore–Penrose Inverse matrix $||A||_F = \sqrt{Tr(AA^T)}$ A have $n \times m$ dimension and have rank $r$ I have ...
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How to deal with the non-uniqueness of SVD in numerical applications?

There are many applications in applied mathematics where the SVD of a matrix comes in handy. For example, consider the problem where we want to find an approximate solution to a(n) (overdetermined) ...
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SVD of a specific matrix and Singular values behaviour

I have a rectangular matrix $A \in \mathcal{M}_{l,n} (\mathbb{C})$, $l>n$ which has this property : $$A=\left[\begin{matrix} M_1 & i M_2 \\ M_2 & -i M_1 \end{matrix}\right]$$ Where $M_i$ ...
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constrained rank approximation

I'm trying to solve a problem similar to this problem. Instead of requiring the diagonals to be 0, I'd like to require columns of the low rank approximation to decrease in value while going down the ...
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Properties of singular value decomposition

Every (real) $m\times n$ matrix $A$ of rank $r$ has an SVD $$ A = U\Sigma V^T $$ Now, I have read about the following properties: $\text{Image}(A) = \text{span}\{u_1,\dots,u_r\}$ $\text{Null space}(...
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Why SVD matrix is orthogonal matrix?

I am trying to understand Why SVD matrix is orthogonal matrix. Let define SVD as A=UDV', I want to prove that U is a orthogonal matrix. I understand each column of U is the eigen vector of AA'. I ...
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Range space $\mathcal{R}(\textbf{A})$ the same as $\mathcal{R}(\textbf{AA}^H)$?

I'm working on a problem as follows: Given $\textbf{A}\in\mathbb{C}^{M\times N}$, show that $\mathcal{R}(\textbf{A})=\mathcal{R}(\textbf{AA}^H)$ where $\mathcal{R}()$ denotes the range space of a ...
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V-matrix in SVD of A

I have some doubts in understanding the concept of SVD. 1) Let's assume I have a matrix $$V^H= \begin{bmatrix} f &e&g&f\\ c &d&d&k\\ m &f&j&e\\ p &l&a&...
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Does the decay rate of singular values reflect linear dependency of vectors in a matrix?

In order to compare two matrices, suppose A and B, based on the level of linear dependency between column vectors of a given matrix, I can think about following measures: Rank of a matrix: If the ...
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why $x = \mathbf A^{\dagger}b$ is the one that minimizes $|x|$ among all mimizers of $|\mathbf Ax - b|$

for arbitrary matrix $\mathbf A\in \mathbb R^{m \times n}$ and $rank(\mathbf A) = r$, solve the least squares: $$\min \|\mathbf Ax - b\|_2. $$ According to SVD, pseudo inverse of $\mathbf A$ is $$\...
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Question about SVD proof from Trefethen and Bau

In Trefethen and Bau's proof of the SVD (see image below), they start by defining the following: $$ U_1^* A V_1 = \begin{bmatrix} \sigma_1 & w^* \\ 0 & B \end{bmatrix} $$ I understand the ...