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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Finding the largest singular value “easily”

Im only interested in finding the largest singular value. I don't need the singular vectors. Is there a way to do so without performing full SVD? Is there an analytical expression? If not, is ...
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SVD: find $\vec{v_2}$ when $\lambda{_2} = 0$ given the relation $A^t \vec{u_2} = \lambda{_2} \vec{v_2}$

An SVD of A given by: Where D is a diagonal matrix containing the singular values $s_1 and s_2$ on the diagonal. Given these related to the SVD of A: $ A = \left(\begin{array}{rrr} -3 & 1 \\ ...
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SVD and least squares solution: orthogonal projection $ \vec{\widehat b} $ of $ \vec{b} $ onto $Col(A^T)$

Given the following: $ A = \left(\begin{array}{rrr} -2 & 3 & 2 \\ 2 & 2 & 3 \end{array}\right).$ A has the SVD: $A = USV^T$ $ b = \left(\begin{array}{rrr} -6\\ 1 \\ 4 ...
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18 views

Simplifying the inverse of the sum of 2 matrices

I would like to simplify the following inverse computation : $$(D+A)^{−1}$$ where $A=UΣU^T$ (eigenvalue decomposition). And D is a diagonal matrix such that $D = \lambda \boldsymbol{I}$ I know the ...
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19 views

inverse of sum of diagonal matrix and eigendecomposition

I would like to simplify the following inverse computation : $$(D + A)^{-1}$$ where $A=U\Sigma U^T$ (eigenvalue decomposition). And D is a diagonal matrix I know the inverse of A is ...
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MatLab Single Value Decomposition [closed]

there is an error in this code that I cannot seem to find. I am trying to take a simple black and white image and preform single value decomposition on it. Here is the code I am trying to use. Link to ...
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Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
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42 views

Find unit singular vectors for two known singular values.

$$A=\begin{pmatrix}-3 & -1 \\ -3& 1\end{pmatrix}$$ Find the singular values of $σ_1$ and $σ_2$, Find unit vectors $v_1$ and $v_2$ such that $||Av_1|| =σ_1$ and $||Av_2|| =σ_2$ I figured ...
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SVD and transpose of a skinny matrix

Show: If $\mathbf{A}\in\mathbb{R}^{M\times N}$ with $M\geq N$, then there exists a matrix $\mathbf{G}$ with orthonormal rows so that $\mathbf{A}^T=\mathbf{G}\mathbf{A}\mathbf{G}$. I'm pretty lost on ...
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how to solve this low rank approximation problem

$$L = \sum_{i=1}^n (N_i-C_i)^T\Sigma^{-1}(N_i-C_i) =Tr((N-C)\Sigma^{-1}(N-C)^T)$$ $$=\sum_{i=1}^n \sum_{j=1}^d (N_{ij}-C_{ij})^2\sigma_j^{-2}$$ $$rank(C)=k<rank(N)$$ Basically i need to find the ...
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Why $||U\Sigma V^\intercal - X_k||_F = ||\Sigma - U^\intercal X_k V ||_F$

Here $U\Sigma V^\intercal $ is the svd decomposition of M.I know that U and V are othogonal but i dont know which property it is using. This in reference to the post Proof of Eckart-Young-Mirsky ...
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minimization of weighted frobenius norm for pca

So my problem is i like to derive pca solution as the maximum likelihood estimate for the true data.So basically i am assuming that my measured data has two component one is low rank component and ...
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1answer
80 views

vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
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34 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
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Compactness argument in SVD existence proof

The classical proof of the existence of the SVD factorization by Trefethen and Bau reports Set $\sigma_1 = \mid\mid A \mid\mid_2$. By a compactness argument, there must be a vector $v_1 \in ...
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Singular value decomposition positive components

I am using Singular Value Decomposition (SVD) applied to Singular Spectrum Analysis (SSA) of a timeseries. ...
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If rows of matrix M are switched, do the singular values of M change?

I have looked for a source on the relationship between elementary matrix row operations and singular values, but I can't find a good, compact set of information. I'm really only interested in ...
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24 views

Efficiently compute the eigenvectors of the Laplacian of a symmetric positive matrix

I am working with a matrix A relatively large (200k x 200k), and I want to compute the eigenvectors of the Laplacian: $L = D - A^2$, where $A$ is symmetric. I don't need all eigenvectors, just a few ...
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51 views

Changes in singular values of matrix when rows are added

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is: Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...
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Significance of minimum singular value

I came across this statement from Wikipedia on singular value decomposition. A total least squares problem refers to determining the vector $x$ which minimizes the 2-norm of a vector $Ax$ under ...
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Singular value decomposition for matrices that are not square?

I understand that the Singular Value Decomposition is defined as SVD = $U\Sigma V^T$ , but I am slightly confused about the calculations when the matrix is not square. For example, I have the matrix: ...
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20 views

how to calculate variance on SVD parameter estimation?

How do i Calculate the variance of a estimated parameter by SVD? I know that there is an uncertainty on the dataset, but how can that be used to calculate variance of an parameter?
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When PSD, singular value is equal to eigenvalue

It is known that If a matrix is PSD (symmetric), then its eigenvalues are equal to its singular value. How to prove it? Hope for a hint. thanks,
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Known the SVD of A, given a diagonal matrix C. How to find $SVD(CA)$?

I have an $M$x$N$ matrix $A$, given a diagonal matrix $C$ is$M$x$M $. What relationship can I find between the SVD decomposition of $A$ and $CA$? I suppose $C$ has no zero entry in the diagonal, so ...
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39 views

Generalized SVD and weighted SVD

I've the following question: How should I select the $A$,$B$ matrices in the generalized singular value decomposition (GSVD) such that it solves the weighted version of the generalized singular value ...
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37 views

If an upper bidiagonal matrix has a repeated singular value, it must have a zero on its diagonal or superdiagonal

I have a question that mentioned in the book "Matrix Computations" by Golub and van Loan. "Show that if $A\in \mathbb{R}^{n\times n}$ is an upper bidiagonal matrix having a repeated singular value, ...
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Is this matrix with SVD diagonalizable

Let $X=U\Sigma V^T$ is an (economical) SVD decompoisition of a square $n \times n$ stochastic matrix $X$, where $U$ and $V$ are two $n \times r$ matrices, and $\Sigma$ is a $r \times r$ matrix. Now ...
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58 views

Do I have to be a mathematician to understand the following papers?

I come from a CS&Machine Learning discipline. I have been looking to understand the core idea of Non-Negative Matrix Factorization. While most of the ML based work is understandable, mostly the ...
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43 views

Multiplying Matrices (using SVD)

I'm an online course and it has sections on it regarding SVD. I understand the concept, however, the maths eludes me. For example, if you view this: you can see the Matrix q * Matrix V = the output ...
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Proof of Optimality for Approximation of Probability Spaces by PCA

I have come across a theorem that states, that the $d$-dimensional subspace found by PCA is the optimal approximation of a probability space with such a plane, in the sense that it minimises the ...
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Gaussian variance estimation via spectral decomposition

I was given a dataset (a mat file) of 100,000 observations, each with 50 dimensions (coordinates). Denote matrix $X$ is a 100,000x50 matrix in which each column was generated according to: ...
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$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
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the relation between positive definite matrix and its ellipsoid

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Then, we associate with each $A \in S^n_{++}$ ...
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In the SVD of $A = U \Sigma V^T$, how does one know that V actually spans the row space $C(A^T)$ of A and U the column space $C(A)$?

In the SVD of $A = U \Sigma V^T$, how does one know that V and U actually span the column and row space of A (respectively for each one)? I do know how to find such a U and V and $\Sigma$ by just ...
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Computing right null vector with smallest eigenvalue

I know that the null space can be calculated from the equation $Ax=0$ by computing $svd(A).$ If we compute svd, $svd(A)=USV,$ $S$ are the singular values, $V$ are the eigenvectors of $A^TA$ and U are ...
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A and P*A*Q have same singular values being P and Q orthonormal matrixes:

Let P and Q be two orthogonal matrices such that it makes sense to calculate PAQ. Show that A and PAQ have the same singular values. So far, I've come to the fact that the SVD of an orthogonal matrix ...
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34 views

How does additive noise change the SVD

For matrix $M$ with SVD $M=U\Sigma V^*$ and random matrix $A$, what is the SVD of $M+A$? That is, how will $A$ change the singular values and vectors of $M$? Let's even say that the entries of $A$ ...
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37 views

Left and right null vectors

Can somebody please explain what is the meaning of left null vector and right null vector? I know that null space of $Ax=0$ (where $x$ is the null vector) can be found out using row echelon form or by ...
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$\text{det}(A+E) = 0 \implies \Vert E\Vert_{2} \geq \sigma_n$?

Suppose $A,E$ are $n\times n$ matrices and $A$ has singular values $\sigma_1\geq \sigma_2 \geq \cdots \geq \sigma_n >0$. Please help me to prove that $\Vert E \Vert_2 \geq \sigma_n$ if $A+E$ is ...
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Matching degrees of freedom of SVD and original matrix

Every real $M\times N$ matrix has at least one singular value decomposition where $U$ is $M\times M$, $V$ is $N\times N$, and $S$ is a $M\times N$ diagonal matrix with at most $\min(M,N)$ non-zero ...
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uniqueness of svd decomposition and its role in statistical analysis

let us consider following model according to following link http://www4.ncsu.edu/~ipsen/REU09/chapter4.pdf it says that : The singular values are unique, but the singular vector matrices are ...
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Singular value decomposition of a matrix multiplied with a semi-unitary matrix

Say I have a $m \times n$ matrix A and I do a singular value decomposition (SVD), $\quad A = U \Sigma V $ where $U$ is $m \times m$ unitary, $\Sigma$ is $m \times n$ with non-zero elements only on ...
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Does the following type of “SVD” exist?

SVD of $A$ gives $U$, $\Sigma$ and $V$ such that $A = U \Sigma V$. I am interested in a different problem. Given an $A$ and $\Sigma_1,\ldots,\Sigma_{n-1}$ diagonal matrices, such that we know that ...
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Tikhonov regularization vs truncated SVD

To find $\mathbf{x}$ such that $$A\mathbf{x}=\mathbf{b}$$ we can use least squares when the problem is not well posed. Further, we can use Tikhonov regularization when $A$ is ill-conditioned. In ...
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If we have SVD of a submatrix, is that useful for SVD of full matrix?

Say we have a matrix that has a singular value decomposition $M_1=U_1\Sigma_1V_1$, and we have another matrix $M_2$, which has the same number of rows. Can we say anything about the SVD of their ...
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Why the largest singular value of a megic matrix is its magic constant?

A magic matrix is a square matrix such that the sums of the elements of each row, each column and diagonal equal to a same number, the magic constant. As reported ...
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If A and A' are approximately the same, are their principal components/SVD very close?

If we have that two matrices $A\approx A'$ within some guaranteed error bound for each term, and $A=U\Sigma V$ is the singular value decomposition for $A$, and $A'=U'\Sigma' V'$ is the SVD for $A'$, ...
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Replacing Singular Values of a Matrix with Complex Ones

Is there a procedure to replace singular values of a real valued matrix according to: s1 -> i*s1 s2 -> i*s2 ... without going through any singular value decomposition (change singular values and ...
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A question in paper “Fitting helices to data by total least squares” writen by Yves Nievergelt in 1996

does anyone read this paper before? I got a problem in this paper. Specifically, I do not understand the Step 2.3 to 2.5. Two variables, r and s, are involved in the calculation but I have no idea ...
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Can a matrix have more than one inverse (Singular Value Decomposition)

Assume there's a matrix $A$ with SVD as below $$ A = U \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ...