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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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Generalized singular value decomposition

I am studying GSVD and trying to understand the concept of an algorithm. As I understood, GSVD possible compute for 2 matrices (GSVD in matlab) Do someone knows how I can compute GSVD for 3 or 4 ...
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Why are singular values of “complex” matrices always real and non-negative?

I've already read the following related questions on math.SE: Why can't singular values be complex numbers? Clarification on the SVD of a complex matrix Why are singular values always non-negative? ...
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Perform Singular Value Decomposition of the following matrix?

Perform Singular Value Decomposition of the following[2*2] matrix and represent in UDV(power)T? [3,2,2; 2,3,-2]
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SVD with non-standard inner product

I have a linear transformation $T: V \to W$ where $V$ and $W$ are finitely generated real inner product spaces and their inner product is not necessarily standard. I also have $K: V \to V$. My goal is ...
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Where does the size of the diagonal matrix in the SVD come from?

SVD says, that we can represent an arbitrary matrix $M$ of size $m \times n$ in the following form: $$ M = U \Sigma V^\dagger $$ Now, I've encountered two verions of this theorem (with different ...
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Singular value decomposition of the Pauli matrix $\sigma_x$

I'm trying to compute the singular value decomposition of the Pauli matrix $$\sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}.$$ According to the SVD theorem, this matrix can be ...
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Singular value decomposition - determinant

I know that when expressing a matrix $A$ in SVD, the determinant of $A$ can be calculated by finding the product of the singular values $\sigma_1 *... *\sigma_n $. My question is, does this hold true ...
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The singular values of the best rank-$k$ approximation to a matrix

Let $A\in\mathbb{C}^{m\times n}$ be a complex matrix. Let $B_k$ be a best rank-$k$ approximation to $A$ such that \begin{equation*} B_k\in\arg\min\limits_{{\rm rank}(B)=k}||A-B||_F, \end{equation*} ...
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What's the effect of normalizing for SVD?

$\mathbf{X} \in \mathbb{R}^{M\times N}$, $M$ is the number of data, $N$ is the dimension of data. Then one can have SVD as $\mathbf{X= U\Sigma V^\top} $. However, now I do the SVD on the tranlsated ...
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Can the derivative of the matrix absolute value explode when we approach singular matrices?

Let $ \text{GL}^+_n$ be the group of real $n \times n$ matrices with positive determinant, and consider the matrix absolute value function, $| \cdot | : \text{GL}^+_n \to \text{Psym}$ given by $|A|=\...
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singular value decomposition of the volterra operator

Can anyone help me to this question? Compute the singular value decomposition of the Volterra operator $Tu(x) = \int_{0}^{x} u (s) ds$ in $L^2(0,1)$ and use it to find $\|T\|$. Is $T$ normal ? ...
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Eigenvalues of a Covariance Matrix with Noise

Imagine to have a covariance matrix $2\times 2$ called $\Sigma^*$. \begin{bmatrix}1+\sigma^2&\rho_{12}\\\rho_{21}&1+\sigma^2\end{bmatrix} I know that $\rho_{12} = \rho_{21}$ because it's ...
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Distribution of column of random orthogonal matrix

Suppose $O \in \mathbb{R}^{n \times r}$ with $r < n$ is a random matrix whose distribution is uniform on the set of $r \times n$ matrices such that $O'O = I_r$. Is is true that the columns of $O$...
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Show that $\min_X \|X\|_2 = \frac{1}{\sigma_1}$

Given $ A\in \mathbb{C}^{m\times n}$ with $\sigma_1$ as biggest singular value, and $\det(I-AX) = 0$ where $ X\in \mathbb{C}^{n\times m} $, can you show that $$\min_X \|X\|_2 = \frac 1{\sigma_1}\:?$$ ...
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When does the geometric limit hold for matrices?

We know that for scalars, $a+ax+ax^2+...\infty=\frac{a}{1-x}$ provided $|x|<1$. Suppose $x$ were a square matrix. What are the necessary conditions for $aI+ax+ax^2+...=a(I-x)^{-1}$ to hold? My ...
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Trace Norm / Nuclear Norm: How to verify?

The nuclear norm is defined by this [from wikipedia]: $$\|A\|_* = \text{trace}(\sqrt{A^*A}) = \sum_{i=i}^{\min\{m,n\}}\sigma_i(A)$$ I get the derivation of this equation. However, I wanted to test ...
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show singular matrix map interior and surface of a unit sphere into an ellipse

The original question image A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, \ldots , z_k)^T$ ...
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Relationships between the singular values and eigenvalues of an asymmetric matrix A

As we know, if A is a real and symmetric matrix, the SVD decomposition is just the eigendecomposition, and the singular values and singular vectors are just eigenvalues and eigenvectors. For the case ...
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Matrix square root of squared correlation matrix

Setup: Given $y \sim N(0,\Sigma)$, suppose we want to transform $y$ to a new space so entries have zero covariance. We can use the inverse square root and apply to transform $\tilde{y} = \Sigma^{-1/...
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Eigendecomposition of $(a^\top X a)X - Xaa^\top X$

Let $a\in\mathbb{R}^n$ be a nonzero vector, $X\in\mathbb{R}^{n\times n}$ be positive definite. What are the eigenvalues and eigenvectors of $(a^\top X a) X - Xaa^\top X$?
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Applications where weighted singular values are useful

Let $A = U\Sigma V^*$ by the compact SVD where rank$(A)=r$ and $\Sigma$ is $r\times r$. If $A$ is Hermitian, then $U=V$. Let us form another matrix $A_k = UK\Sigma V^*$, where $K\ne I$ is positive ...
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Can we choose smoothly the singular vectors of a matrix?

Let $X$ be the space of all real $n \times n$ matrices, with strictly negative determinant, and pairwise distinct singular values. $X$ is an open subset of the space of all real square matrices. Is ...
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Studying series of oscillating matrices via SVD

I have a series of matrices, very big matrices of size ~ $4000 \times 4000$, \begin{equation} A = \{A_1, A_2, ...A_n\}, \end{equation} and the series oscillates, meaning if I take Fourier transform ...
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truncation degree of decomposed covariance matrix

I have a covariance matrix of a standardized data set. Doing a singular value decomposition i find near zero singular values and would therefore like to truncate it. I know of Picard plots which ...
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How does the sign of a p.s.d matrix entry change if we reconstruct the matrix with low-rank SVD?

Given a matrix symmetric, p.s.d matrix $K$, and all the entries of $K$ are positive. $J$ is a low-rank approximation of $K$ following the SVD procedure (the one Eckart–Young–Mirsky theorem describes)....
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Rows of V in reduced SVD with norm 1

Suppose, we're given the reduced/compact SVD of the rank-$r$ Matrix $A=USV^T$ where $U\in\mathbb{R}^{m\times r}$, $S\in\mathbb{R}^{r\times r}$ and $V\in\mathbb{R}^{n\times r}$ and suppose the $i$-...
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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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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 ...