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Questions tagged [singularvalues]

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Spectral norm of matrices with complex eigenvalues

Suppose that $M$ is a square, invertible matrix with eigenvalues $\lambda_1, \ldots, \lambda_n$ where the lambda's can possibly be complex. Suppose that $\lambda_{\max}(M)$ is complex valued. How 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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Smallest singular value of a specific structured matrix

Consider the matrix $$ A = \begin{bmatrix} 1 & \alpha_1 \\ 1 & \alpha_2 \\ \vdots & \vdots \\ 1 & \alpha_n\end{bmatrix} $$ where $\alpha_1, \dots, \alpha_n$ are real numbers. I was ...
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Solve $\mathop{\arg\max}_{{v \in \mathbb{R}^m, \| v \| = 1}} v^T A A^T v$ with SVD

Let $A \in \mathbb{R}^{m \times n}$ be a matrix with full rank and $m \le n$. How can we solve the problem $$ \mathop{\arg\max}\limits_{\substack{v \in \mathbb{R}^m \\ \| v \| = 1}} v^T A A^T v $$ ...
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Largest solution of a linear system

Given an $n\times m$ matrix $A$ of full-column rank, and a vector $\vec b$ of size $n$. We consider the solution of the linear system: $$ A\vec{x}=\vec{b} $$ Since $A$ is full-column rank, the ...
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1answer
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Higher order Inverse Function Theorem

Consider a smooth function $F : \Bbb{R}^n \to \Bbb{R}^m$ where $n \geq m$ are positive integers. Consider a curve $\alpha : [0,1 ) \to F^{-1}(0)$ such that $\alpha(0) = x \in F^{-1}(0)$. Then $F(\...
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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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Perturbation bound on approximate linear system

Suppose that $A, \hat A$ are invertible real matrices such that $Ax = y$ and $\hat A\hat x = \hat y$, where $\|A - \hat A\| \leq \epsilon_1$ and $\|\hat y - y\| \leq \epsilon_2 \|y\|$. I'm trying to ...
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2answers
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Bounding the coefficients of the characteristic polynomial of a matrix

For a given $n \times n$-matrix $A$, the characteristic polynomial of $A$ is $\lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0$. I am curious to know if we can upper bound the coefficients of this ...
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$\| M - A \|_F^2 \geq \sum_{h = l+1}^k \lambda_h$

Let $A \in M_{n,p}(\mathbb{R})$ such that the rank of $A$ is $k \leq \min(n,p)$. Moreover let $M \in M_{n,p}(\mathbb{R})$ be a matrix of rank $l$ with $l+1 \leq k$. We denote $(\lambda_i)_{i \leq k}$ ...
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SVD of a $3\times 3$ matrix

I am trying to implement an efficient, ad-hoc method to perform the SVD of a $3\times3$ matrix. I know how to obtain the Eigen values of such a matrix, by direct resolution of the characteristic ...
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32 views

When Singular Value and Eigenvalue are coincide

For which matrix Singular Value and Eigenvalue are coincide? I found this question about it, but is their any definition for all the the matrix in with this group
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A good formula for singular value matrix of SVD?

For the SVD, $A_{m\times n}=U_m\Sigma_{m\times n}V^T_n$ where $U$ & $V$ are orthogonal matrices & $\Sigma$ is diagonal, I am trying to obtain a formula for $\Sigma$... If $\Sigma\Sigma^T$ was ...
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1answer
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Why do $M^\dagger M$ amd $MM^{\dagger}$ have the same first $k$ eigenvalues?

In the derivation of the singular value decomposition, it is stated without proof in my notes that there is a relationship between the singular values of $M$ and the eigenvalues of $M^\dagger M$ amd $...
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Characterization of singular values

Let $A \in \mathbb R^{m\times n}$ and its singular values be denoted by $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_n \geq 0$. Then $$ \sigma_i(A) = \min\{\|B \|_2\colon B \in \mathbb R^{m \times n}, \...
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Singular value inequality for sum of 2 matrices

I found a theorem mentioned in a couple of places, but could not find a proof. The theorem states the following: Let $A, B \in \mathbb{F^{m,n}}$, $p=min(m,n)$ with singular values $\sigma_1(A) \...
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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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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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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
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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
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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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1answer
30 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
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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 ...
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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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1answer
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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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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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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
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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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1answer
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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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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
34 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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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 $U \in \mathbb C^{d \...
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Relationships between top-$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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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
155 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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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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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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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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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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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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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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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 $...