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Questions tagged [singularvalues]

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If A is a nxn singular matrix, then it has a singular value = 0

This is a question on a testexam. But am I correct in assuming that a singular matrix has det = 0, which gives it an eigenvalue of 0 and that gives it a singular value of 0?
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Do I get better solution if I have more data - Pseudo Inverse

I wonder if I can get a better solution for this equation: $$Ax = b$$ If $A$ is not square and I use pseudo inverse $A^{\dagger}$ to find $x$ $$x = A^{\dagger}b$$ The reason why I asking this is ...
3
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1answer
56 views

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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1answer
24 views

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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2answers
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Is the set of matrices with distinct singular values connected and dense?

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$ Consider the set $$ X=\{ A \in \GLp \,\,|\,\, \text{ all the singular values of } A \text{ are distinct } \},$$ where ...
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1answer
31 views

How do I find the numbers of large population? Statistics

I have a vector $$[10000, 1000, 800, 700, 500, 100, 12, 12, 12, 11, 8 , 7,6,4,3,1,0]$$ And I want to find out how many large numbers there are in my vector, which I call my population. In this case,...
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1answer
32 views

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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1answer
22 views

Absolute of all eigenvalues are always bounded by maximal singular value

There are many discussions about the singular values and eigenvalues, such as What is the difference between Singular Value and Eigenvalue?. I want to ask the particular one in title. Usually, for a ...
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0answers
50 views

Does convergence in Hilbert-Schmidt norm imply convergence of singular values?

Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = \sum_n \lambda_n u_n \otimes v_n$. Now let $A_i$ be a sequence of operators ...
2
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0answers
32 views

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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1answer
28 views

Singular and eigen values properties…

Let $A\in\mathcal {M}_n(\mathbb{R})$, we will denote $\lambda_{\max}(A)$ the biggest eigenvalue of $A$ in absolute value, as for $B\in\mathcal M_{m,n}(\mathbb{R})$ we will denote $\sigma_{\max}(B)$ ...
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0answers
29 views

Eigen-decomposition of real Symmetric psd matrix

I am doing PCA analysis on a large amount of data and I need to compute the eigen-decomposition of the covariance matrix which is a real, symmetric, positive semi definite matrix. It doesn't get any ...
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0answers
24 views

Relationship between least singular value and exact cover of image

Is the following fact true? If so, how would one go about providing a proof or if not then disproving? The least singular value of a matrix determines exactly the radius of the largest ball that ...
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1answer
22 views

Identity between resolvent and singular value density

I was reading the paper Sengupta, Anirvan M., and Partha P. Mitra. "Distributions of singular values for some random matrices." Physical Review E 60.3 (1999): 3389. but got stuck at equation (3):...
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Bounds on singular values of invertible 0-1 matrices

I'm interested in considering digraphs from an algebraic perspective, which leads me to the following question. Consider an invertible 0-1 matrix of shape $n \times n$. What lower and upper bounds ...
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8 views

properties of singular values of a complex matrix

Suppose $G$ is a complex $n\times n$ matrix Could anyone help me to prove the following where $\sigma$'s are singular values of $G$? $\det G\ne 0 \Leftrightarrow \sigma_{\min}[G]>0$. $\sigma_{\max}...
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Given the $SVD$ of a matrix comprised of centered data points in $R$3, how do I find the line of best fit through the origin?

I'm slightly confused by this question - from my understanding the vectors u in $U$ are first principal components. In that case, would they already compose the line of best fit through the origin (...
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0answers
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Maximum Singular Value of $\textbf{A} -\textbf{B}$ for a Certain $\textbf{B}$

Let $\textbf{A} \in \mathbb{C}^{n \times n} $, such that $rank(\textbf{A}) = r$ and the singular values of $\textbf{A}$ be $\sigma_{1} \geq \dots \geq \sigma_{r} > 0$. Let $\textbf{u}_j$ and $\...
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1answer
33 views

Singular vectors of a symmetric block secondary diagonal matrix

Given $A \in \mathbb{R}^{n \times m}$, consider the symmetric matrix $M = \begin{pmatrix} 0 & A \\ A^{t} & 0 \end{pmatrix} \in \mathbb{R}^{(n+m) \times (n+m)}$. Show that a simple ...
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28 views

Schur factorization from spectral decomposition

Let $A \in \mathbb C^{d \times d}$. The Schur factorization of $A$ is the decomposition $A = U T U^\ast$, where $T \in \mathbb C^{d \times d}$ is an upper triangular matrix and $U \in \mathbb C^{d \...
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0answers
32 views

Relationships between top-$k$ eigenvector and top-$k$ singular vector of symmetric matrix $A$

Is there any relationships of top-$k$ eigenvector and singular vector of symmetric matrix $A \in R^{n \times n}$? For symmetric matrix $A$ its eigenvalue decomposition is: $$ A = B \Lambda B^T$$ ...
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71 views

Condition number of augmented matrix

I am trying to solve following problem. Let $A$ be a $m$ by $n$ ($m\geq n$) full rank matrix. What is then condition number of its augmentation $M$: $$ M = \begin{bmatrix} I & A \\ A^{\top} &...
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1answer
105 views

Summation of singular values

If $A \in \mathbb{R}^{m\times n}$, then show that $$\sum_{s=1}^{r}\sigma_s(A)=\text{max}\{\text{trace}(U^TAV): U \in \mathbb{R}^{m\times r}, V\in \mathbb{R}^{n\times r}, \text{and} \ U^TU=V^TV=I_r\}.$$...
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33 views

Confidence Intervals for a Line fit in 3D using SVD

I have a set of points in 3D and I have performed singular value decomposition on in order to determine a line of best fit (see https://stackoverflow.com/questions/2298390/fitting-a-line-in-3d). I ...
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0answers
89 views

Prove polynomial on $\mathbb{CP}^1$ has finite singular values

Suppose I have a polynoial $p:\mathbb{C} \to \mathbb{C}$. There's a natural way to extend this polynomial to a polynomial $p:\mathbb{CP}^1 \to \mathbb{CP}^1$. If you have a polynomial $p(z) = a_nz^n + ...
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0answers
40 views

relationship between the sum of a matrix A's singular values and $max[trace(U^TAV)]$

For a $m \times n$ matrix $A$, how to show $\Sigma_{k=1}^{r}\sigma_k(A) = max\{trace(U^T A V)\}$, where $U$ is m by r, V is n by r, and $U^TU = V^TV = I$? ( $\sigma_k(A)$ are the singular values of A )...
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SVD with operators

If $T(p)=p'$; where $p$ is 3rd degree polynomial and $p'$ is its derivative, and inner product is defined as $\langle p,q\rangle = \sum_{i=0}^{n} \alpha_i \beta_i$; where $\alpha_i, \beta_i$ are ...
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Skew-Symmetric singular value problem.

Need a start on the following problem, see image if the text version below is hard to read. Let A be skew-symmetric, and denote its singular values by σ1 ≥ σ2 ≥···σn ≥ 0. Show that (a) If n is even,...
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1answer
109 views

Evaluating $K\big(\frac{3-\sqrt{7}}{4\sqrt{2}}\big)$

On MSE, I have seen derivations of the elliptic integral special values $$K(1/\sqrt{2})=\frac{\Gamma^2(1/4)}{4\sqrt{\pi}}$$ $$K(\tan(\pi/8))=\frac{\sqrt{\sqrt{2} +1} \Gamma (1/8)\Gamma (3/8)}{2^{13/4}\...
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2answers
39 views

U matrix in Singular value decompositon.

I know that the Singular Value Decomposition of a matrix $X$ is given by: $X = U\Sigma V^T$, where $U$ and $V$ matrices are column orthonormal and $\Sigma$ is a diagonal square matrix containing ...
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1answer
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If $A= (QU_2)\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix} (PV_2)^T$, When will diagonal elements of $\Omega_K, \Lambda$ be singular values?

Suppose we show $$A= (QU_2)\begin{bmatrix}\Omega_k & O\\ O& \Lambda\end{bmatrix} (PV_2)^T$$ where $\Omega_k$ and $\Lambda$ are the diagonal matrices from the SVD of two other matrices. When ...
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0answers
57 views

nocedal and wright singular values bounded away from zero

In the Nocedal and Wright Numerical optimization second edition book, pages 255-256, they state that the Jacobians "$J(x)$ have their singular values uniformly bounded away from zero in the region of ...
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30 views

uniqueness of pseudoinverse

I am reading Linear Algebra by Friedberg. In page 414 chapter 6.7, the first paragraph of the section of pseudoinverse of a matrix writes Let $A$ be an $m\times n$ matrix. Then there exists a unique $...
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0answers
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Alternate definition of “SEPARABLE MATRIX”

I am familiar with singular value decomposition based definition of separable matrix which counts on the number of non-zero singular values. If there is only one non-zero $\sigma$, matrix is rank-1/...
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1answer
79 views

Comparing SVD of orthogonal matrix $(\sqrt{W}A)$ and matrix $A$ in order to find the singular values of A

I have a diagonal matrix W and another matrix A. I also have that $(\sqrt{W}A)$ is an orthogonal matrix. I have that the SVD of $$\sqrt{W}A=U\Sigma V^T=(\sqrt{W}A)II$$ where I is the identity matrix....
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1answer
59 views

Canonical angle between two subspaces and singular values

I saw the statement "Given two subspaces $\mathbf{V}_1$, $\mathbf{V}_2 \subseteq \Re^d$, then the canonical angles between the subspaces $\mathbf{V}_1$ and $\mathbf{V}_2$ is given by the singular ...
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1answer
24 views

Why do the singular values of $A$ scale by $\sqrt{k}$ as I compute the SVD on $k$ copies of $A$?

This is a research question I have been stuck on for some time. Consider $U\Sigma V^T = A$. Suppose I compute $A_2 = [A;A] = U_2 \Sigma_2 V_2^T$, then $\Sigma_2 = \sqrt{2}\cdot\Sigma$. Similarly, if ...
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1answer
68 views

Eigenvalues and singular values of $3 \times 3$

I'm asked to provide an example or disprove the existence of a $3\times 3$ matrix whose eigenvalues are all $0$ but has singular values $1$, $1/2$, $0$. My initial instinct was to consider the ...
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second largest singular value of a doubly-stochastic irreducible primitive matrix

For an irreducible primitive stochastic matrix $A$, $1$ is the largest eigenvalue and is simple, then the second largest eigenvalue has module strictly smaller than $1$. I also know that the largest ...
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2answers
90 views

Find a singular value decomposition of the matrix

Fina a singular value decomposition for $$A=\begin{bmatrix} -2 & 2 \\ -1 & 1 \\ 2 & -2\end{bmatrix}.$$ Find a (full svd)singular value decomposition for the matrix $A$. My working ...
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0answers
39 views

Norm of difference of matrices

The exercise says: Matrices $A, B\in\mathbb{R^{m\times n}}$ have the same rank $r$ where $r\leq n\leq m$. Prove that $$\sigma_r^A - \sigma_r^B\leq ||A - B||_2\text{,}$$ where $\sigma_i^C$ is the $i$-...
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0answers
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Relating the eigenvalues of two positive semi-definite matrices

Let $A$ be an $n \times n$ positive semi-definite matrix, and let $B$ be an $m \times n$ matrix. Can one say anything relating the eigenvalues of $M_1 := BAB^\top$ and $M_2 := BA^2 B^\top$? In ...
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1answer
66 views

Can we characterize the equivalence classes of matrices up to left multiplication by an orthogonal matrix?

Let $M_n$ be the space of $n \times n$ real matrices, and consider the following equivalence relation on $M_n$: $A \sim B$ if there exist $Q \in O(n)$ such that $A=QB$. Can we characterise nicely ...
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2answers
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If $|A_{ij}|\leq 1$ for all entries of symmetric $A$, then the largest possible spectral norm of $A$ is smaller than $n$?

In other words, for any symmetric or antisymmetric matrix $A\in\mathbb{R}^{n\times n}$ that has $|A_{ij}|\leq 1$, can we conclude that \begin{align*} \|A\| \leq \|J\|, \end{align*} where $\|\cdot\|$ ...
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1answer
65 views

The spectral norm of real matrices with positive entries is increasing in its entries?

Suppose that I restrict myself to $M_{n \times n}(\mathbb{R}_+)$ the set of real matrices with positive entries that are square and have size $n$, and I denote by $\|\cdot\|_2$ the spectral norm of ...
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1answer
32 views

Lower bound on largest singular value of a matrix

I was generating random matrices in Mathematica and noticed that the largest singular value $\sigma_\max$ is always greater than the largest entry of the matrix. I.e., for a matrix $A=\left(a_{ij}\...
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1answer
35 views

To prove that $\vec{u_3}$ is a left singular vector of $A$

Question: Let $A$ be a $3*2$ matrix with $rank(A)=2$. Let $\sigma_1$ and $\sigma_2$ denote the singular values of $A$. Let {$\vec{v_1},\vec{v_2}$} be an orthonormal basis for $R^2$ of right singular ...
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1answer
40 views

Finding the maximum value of $A\vec{x}$ when $||\vec{x}||= 1$

Question: If $A= \begin{bmatrix} 1&0&1\\1&1&-1 \end{bmatrix}$ and $\vec{x}$ is an element of $R^3$. It is given that $||\vec{x}|| =1$. I have to find the maximum value for $A\vec{x}$. ...
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1answer
246 views

How SVD for the frobenius norm has been calculated?

![From the paper for Generalized low-rank y Stephen Boyd, this Frobenius loss function has been used using SVD. Can someone explain it to me the following equation? Is U inverse is equal to U ...