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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Identifying indeterminable terms in polynomial fit

I am using SVD to fit a polynomial surface to a set of points, where the number of points may be less than, equal to, or more than the number of polynomial terms. For simplicity, let's assume points ...
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SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
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Null Bilinear Forms $x^T A y = 0$, where $A$ is square and full rank.

Let A be a full rank square matrix (A has no null space). When does $y^T A x = 0$ occur ? It could be that this problem is case-specific, so please find attached a document where x,y, and A take ...
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Invariants of a real symmetric matrix

Problem. I have a real symmetric $n \times n$ matrix $A$ and would like to compute a set of real numbers $f(A) = (x_1, \ldots, x_m) \in \mathbb R^m$ which are invariant under multiplication of $A$ ...
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22 views

Spectral decomposition matrix proof

Suppose we have a square matrix $A$ with singular value decomposition $A=U\Sigma V'$. How can we show the equation $$\begin{bmatrix}0 & A^T \\ A & 0 \end{bmatrix}=\frac{1}{\sqrt{2}} ...
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2-norm of the orthogonal projection

So far, I've deduced that if the rank of A is n, then all the columns of A are linearly independent since A has n columns. As a result, m must be greater than or equal to n. In the case that m = n, ...
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QR(pivot) vs SVD for low rank approximation

Define the low rank problem as finding the approximation of matrix A, B: where we want to minimize rank(B) and we want the 2 norm of the residu of A-B to be less than epsilon. Could someone help me ...
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How to take the SVD of this “almost” moore-penrose inverse?

I have an expression of the form $(W^TAW)^{-1}W^T$, where $W \in \mathbb{R}^{m\times n}$ and $A\in \mathbb{R}^{m\times m}$ and $A$ is positive semidefinite. I would like to take the SVD of this ...
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23 views

How is it the matrix-vector multiplication with SVD is $O(m+n)$?

Assuming the singular value decomposition is known, how is it the matrix-vector product of $\mathbf{A}$ ($m \times n$) and vector $\mathbf{x}$ ($n \times 1$) has $O(m+n)$ complexity? Somewhat related: ...
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Can we obtain a Hadamard matrix as sum of two matrices of decaying singular values?

Is there a kind of matrices in $[-1,1]^{n\times n}$ that, when we sum two of them, we obtain a Hadamard matrix of order $n$ (assumed that $n$ is $1$, $2$ or divisible by $4$) as their sum? For ...
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Decompose matrix into directions with uniform variance

Singular Values Decomposition (SVD) can be viewed as decomposing a matrix $M\in\mathbb{R}^{N\times M}$ into directions such that for any $k=1..N$ the k first directions capture the largest amount of ...
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Distribution of Singular Vectors and Values

Let $X$ be a real-valued, full rank $m \times n$ matrix with probability distribution absolutely continuous with respect to the $mn$-dimensional Lebesgue measure. $X$ can be represented by its ...
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1answer
21 views

Can 3 transformations (V, Σ, U) of SVD to describe a perspective transformation?

As known SVD (Singular value decomposition) is a factorization of the form M = UΣV∗. https://en.wikipedia.org/wiki/Singular_value_decomposition SVD of the linear map T can be easily analysed as a ...
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What does the orthogonalisation step for U and V in SVD do exactly?

I'm reading about Singular Value Decomposition and I'm confused about one aspect. I hope someone can clarify this for me. So the SVD looks like this: $C = U \Sigma V^T$. One always notes something ...
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approximations for largest eigenvector for matrix with very high correlations

I was wondering if there are any approximations to calculate the largest eigenvector / eigenvalue for a correlation matrix with high correlated off-diagonal elements (e.g. correlations > 0.8), without ...
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1answer
53 views

Projection of a real symmetric matrix to a cone (face)

Hi Here is my question: Suppose there is an $n \times n$ real symmetric matrix $X$. It is easy to project it onto the positive semidefinite cone $\mathcal{S}_n^+$. We can just apply the the ...
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How to get the SVD of $2AA^T-diag(AA^T)$ given $A$ and its SVD $A=USV^T$?

Given a matrix $A\in R^{n\times d}$ with $n>d$, and we can have some fast ways to (approximately) calculate the SVD (Singular Value Decomposition) of $A$, saying $A=USV^T$ and $V\in R^{d\times d}$. ...
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Modifying the result matrices of SVD so that $\Sigma$ is canonical form

I have the following matrix: $ C'^* = \left( \begin{array}{ccc} -0.0045 & -0.0059 & 0 \\ -0.0059 & -1.0000 & 0 \\ 0 & 0 & 0 \end{array} \right)$ I put it through the SVD ...
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When can the result from SVD and Eigen Decomposition will be same?

When can the result from SVD and Eigen Decomposition will be same? Can we obtain results of one from another? Karthik has suggest a hint here, but has not told how to? Algebraic pavel has suggested ...
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55 views

Image of unit sphere being hyper ellipse proof (SVD)

When I check for the proof of singular value decomposition, they all assume the following is true: The image of the unit sphere under any $m * n$ matrix is a hyper ellipse. However I could not ...
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78 views

why S in SVD is a vector instead of a matrix?

I know that when applying SVD on a matrix (m * n) I should have these three outputs: ...
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1answer
63 views

Proof of “Singular values of a normal matrix are the absolute values of its eigenvalues”

I want a simple proof of this fact using only definitions and basic facts. I've searched for it for some time and I couldn't find a satisfying proof. So I attempted to do it myself. Let $A \in ...
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Pseudo inverse of a singular value decomposition SVD is equal to its “real” inverse for a square matrix?

I was reading this book on numeric linear algebra and it said pseudo inverse of a singular value decomposition (SVD) is equal to it's "real" inverse for a square matrix. It said it is quite clear that ...
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Tracking Matrix Decomposition after row subset selection

Imagine we have $X$, a $n\times m$ non-negative matrix. We take the rank-r SVD of X $$ X = U\Sigma V^T$$ I'm now interested in knowing the decomposition of $X_2$, a $n_2 \times m$ matrix formed by a ...
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What is the best way to compute Pseudoinverse of a matrix?

Mathematica gives the pseudo-inverse of a matrix almost instantaneously, so I suspect it is calculating the pseudo-inverse of a matrix not by doing singular value decomposition. Since the ...
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1answer
30 views

Moore-Penrose inverse solves the Least square solution

The normal form $ (A'A)x = A'b$ gives a solution to the least square problem. When $A$ has full rank $x = (A'A)^{-1}A'b$ is the least square solution. How can we show that the moore-penrose solves ...
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Why do the singular vectors $u_i$ and $v_i$ become oscillatory as $\sigma_i$ decreases?

I am confused with the statement "The characteristic feature of the SVD promises that as $\sigma_i$ decreases, the singular vectors $u_i$ and $v_i$ become more and more oscillatory", which is in my ...
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Intuition for the singular vectors and the outer products in SVD of a non-diagonalizable matrix

I'm trying to better my understanding of the singular value decomposition of an $m \times n$ matrix $M$ of rank $r$. Using the same notation as wikipedia: $$M = U \Sigma V^*$$ where $U$ is $m \times ...
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SVD of concatenate matrices

Let $X \in \mathbf{R}^{n\times m}$ be split into two sub matrices, $X = \begin{bmatrix} X_0 \\ X_1 \end{bmatrix} $, if $X_0 = U_0 \Sigma_0 V_0^T$ and $X_1 = U_1 \Sigma_1 V_1^T$ can we say ...
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Solve the problem $Ax = 0$ when $A$ has full rank.

Generally, the answers $x$ of this least square problem $$Ax = 0$$ where $A = []_{m\times n}$ and $x = []_{n\times 1}$ are in the null space of $A$. I know that people usually use the right-most ...
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1answer
40 views

Largest eigenvector as the rank-1 approximation

Consider a square real symmetric matrix $A$. The singular value decomposition (SVD) is given as $A = U\Sigma U^T$, and it can also be found by minimizing $|A-U\Sigma U^T|$ where $|.|$ is the $l$2 ...
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How to measure matrix separability?

I am looking for ideas to address the following problem. I am dealing with matrices $\mathbf{A}_k\in \mathbb{R}^{n\times m}$ for $k\in \mathbb{Z^+}$. From the notion of matrix separability, we can ...
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Deriving Singular Value Decompositionusing diagonalizability

I'm trying to prove that a real matrix $A$ can be decomposed as $A = U \Sigma V^T$, where $V$ has orthogonal eigenvalues of $A^TA$ along its columns, $\Sigma$ contains the square roots of the ...
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27 views

SVD on a sparse matrix

I am working on a machine learning classification problem. My data matrix is very wide and I am trying to understand the situation I'm dealing with on a deeper theoretical level, not just to use ...
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Effect of the Tikhonov regularization on the Least Squares solution.

Consider the system $Ax = b$, where $b \notin \mathcal{R}(A)$. Then we look to find a least squares solution to this problem via $$ x_{LS} =A^{\dagger}b $$ where $A^{\dagger} = V \Sigma^{\dagger}U^*$ ...
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Comparing the orders of complexity.

I do have to evaluate a property defined as $$w = \sqrt{\det\left(\textbf{J}\,\textbf{J}^\text{T}\right)} = \prod_{i = 1}^n \sigma_i,$$ where $\textbf{J} \in R^{3 x n}$ and $n>3$. Using the ...
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Can you obtain the QR decomposition through SVD?

Is there a straightforward connection between the QR and SVD decompositions of a matrix? To make the question more precise, if I have machine that can compute the SVD of a matrix, but not the QR, ...
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26 views

Singular value decomposition when eliminate columns

I am trying to develop an expression for the singular value decomposition of a matrix as I eliminate columns. Suppose I have a matrix $M_{3}$, which for simplicity has 3 columns. This matrix has the ...
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Given the SVD decomposition of a matrix A, how do I find a basis for col A?

If I have a matrix A whose SVD decomposition is ...
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114 views

Why is the transformation not unique if eigen values are repeated or zero

I am using the formula here to computer transformation between two co-ordinate systems in my 3D game (2 sets of same number of points with co-relation). ...
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1answer
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How to calculate the single value decomposition. when eigenvalue=0?

After calculating the eigenvalues. I got 0 as one of my 2 eigenvalues. Hence, one of the singular values of my matrix is 0. So $\sigma_1 = 0$ But then when Im trying to find U: $$u_1 = ...
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Compare the $k$-th singular values between $MN$ and $N$

Let $M$ and $N$ be two symmetric positive definite matrices with eigenvalues $\geq 1$. Prove that the $k$-th singular value of $MN$ is greater than or equal to the $k$-th singular value of $N$.
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Why are the ordered singular values of $A^{−1}$ given by $\sigma _n^{ - 1} \ge … \ge \sigma _1^{ - 1}$?

Let $A\in{M_n}$ be nonsingular, with ordered singular values $\sigma _n^{} \le ..... \le \sigma _1^{}$. Is it true that the ordered singular values of $A^{−1}$ are $\sigma _n^{ - 1} \ge ..... \ge ...
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Orthogonal projectors and SVD

Let $w_1,...,w_r ∈ C^n$ be arbitrary orthogonal vectors. The orthogonal projector onto the complement of the subspace spanned by the $w_i$ is $P = I − \sum_{i=1}^{r} \frac{w_iw_i^*}{w_i^{*}w_i}$. ...
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Computing columns of a pseudo-inverse

I need to compute the pseudo-inverse of a very large rectangular dense matrix without any special structure or properties. I run out of memory/computing power and have no access to a large parallel ...
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41 views

Matrix values increasing after SVD, singular value decomposition

I am trying to learn SVD for image processing... like compression. My approach: get image as BufferedImage using ImageIO... get RGB values and use them to get the equivalent grayscale value (which ...
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1answer
45 views

What is the SVD of $ab^T+ba^T$?

If $a$ and $b$ are column vectors of equal dimension, is there an analytic formula for the SVD of $ab^T+ba^T$? From a few trials I ran with Mathematica, it appears that there are only two non-zero ...
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Is there an iterative procedure that pushes the singular values of a matrix toward unity?

Consider a square matrix $A = U \Sigma V^T$. I want to find $B = UV^T$ — however, it seems wasteful to compute the whole SVD just to re-multiply the two orthogonal matrices. Does there exist some ...
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Can we predict the number of non-zero singular values in this case?

If there are two matrices $P$ (dimensioned $m\times 1$) and $Q$ ($n\times1$) and a matrix $M$ is constructed by $M=PQ'$ (where the ' indicates transpose), so $M$ is of size $m\times n$. Does $M$ ...
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Extracting information from Singular Value Decomposition.

I am currently working on a heat pump system. The problem involves multiple inputs and outputs. During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to ...