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Questions tagged [singular-values]

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SVD decomposition of a matrix

I have a question regarding singular value decomposition of a matrix which I have not been able to answer. Why the matrices in the svd decomposition of a matrix, are one rank matrices?
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10 views

Maximum Singular Value of Sum of Positive Semidefinite Matrices

We have two real matrices $A$ and $B$. Let $\sigma_{\max}(A)$ and $\sigma_{\max}(B)$ denote the maximum singular value of matrices $A$ and $B$, respectively. Intuitively, the maximum singular value of ...
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When are the singular functions in the singular system of a compact operator in L2 bounded in sup norm?

Suppose I have some compact linear operator $A: L_2 (\mu_1)\to L_2(\mu_2)$ where $\mu_1$ and $\mu_2$ are probability measures on some subset of Euclidean space. Such an operator admits a singular ...
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1answer
36 views

when singular value decomposition is equal to eigenvalue decomposition

I've read in my textbook that the right singular vector $v_i$ is actually the eigenvector of $A^TA$ with eigenvalue $\sigma_i^2$, and the left one $u_i$ is the eigenvector of $AA^T$.So I guess if A is ...
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1answer
60 views

Relationship between singular values of $A$ and eigenvalues of $B:= \begin{bmatrix} 0 & A \\ A^\ast & 0 \end{bmatrix}$

Let $$B:= \begin{bmatrix} O_m & A \\ A^\ast & O_n \end{bmatrix}$$ where $A$ is an $m \times n$ matrix. Find the relationship between the two: Singular values and singular vectors of $A$...
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Why $\Sigma$ in the SVD of a matrix $A$ is invertible? [closed]

I know singular value decomposition of a matrix $A$ is $A=U\Sigma(V^T)$. And if we find to $x$ in $Ax=b$ we can use $A=U\Sigma V^T$ to get $x=V\Sigma'(U^T)b$. But could somebody explain why $\Sigma$ ...
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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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Questions about singular value decomposition

I'm still having trouble understanding the separation between diagonalization and singular value decomposition. I am working with complex matrices so for an arbitrary matrix, I am able to write it as $...
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27 views

Singular Values for Gaussian random matrices

Suppose $X$ is random matrix with rows ${X_1^T, X_2^T, \ldots X_N^T}$ where each of the $X_i$ is an independent random vector with the Gaussian distribution $\mathcal{N}(0,K)$ ($K$ being the ...
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SVD Of Product of Permuted Matrices

Let $A$ be a $n \times p$ matrix and $B$ be $p \times m$ with values in $\mathbb{C}$. Let $C$ be a $n \times p$ matrix whose entries are a permutation of the entries of A and let $D$ be a $p \times ...
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1answer
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If the singular values of an $n{\times}n$ matrix $A$ are all $1$, is $A$ necessarily orthogonal?

Exercise 33 of Section 8.3, from Linear Algebra with Applications (5e) by Otto Bretscher: If the singular values of an $n{\times}n$ matrix $A$ are all $1$, is $A$ necessarily orthogonal? I ...
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Singular Values of an $n$ dimensional matrix

I am interested in calculating the maximum singular value of an $n$ dimensional matrix in python. Can I reshape this matrix as a $2$-D matrix and calculate its operator norm? Are the singular values ...
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1answer
22 views

Change the coefficients in matrix decomposition

Part 1: Suppose $A=\sum_{i=1}^n\sigma_i v_i v_i^\top$, where $\sigma\ge 0$ is a non-negative real number (singular value) and $v_i\in\mathbb{R}^n$ is a vector. Suppose $A$ is positive semidefinite. ...
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Given A non-singular, find E with minimal \sigma_max(E) such that A+E is singular

As given in the title, there is a matrix given, namely: $A = \begin{bmatrix}-1 & 0& 0& 0\\1& -1& 0& 0\\1& 1& -1& 0\\1& 1& 1& -1\end{bmatrix}$ ...
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What is the SVD of the 1x1 matrix with entry $-1$?

Question as the title. Consider the $1 \times 1$ matrix $M$ whose single entry is $M_{11} = -1$. What are the factors in the SVD of $M$? Keep in mind that singular values are non-negative by ...
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1answer
34 views

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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1answer
44 views

If $A \in C^{nxn}$ , $A \ge 0 $ and A is sing., there exists a sequence of matrices $C_k$, that $C_k \ge 0$,$|C_k| = 1$ and trace $AC_k \le 1/k$

Question: Show that if $A \in C^{n \times n}$ , $ A \ge 0 $ and A is singular, then there exists a sequence of matrices $C_k$, $k = 1,2,...$ such that $C_k \ge 0$, det $C_k = 1$ and trace $AC_k \le 1/...
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Can we perturb a map to have distinct singular values?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
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Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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Perturbation Bound for Left Singular Matrix

Let $A$ and $\hat{A}$ be two $N\times N$ matrix with rank r, let $A=\hat{A}+H$ where $H$ is some perturbation. Suppose $A$ and $\hat{A}$ have following SVD: \begin{align} A&=U\Sigma V^T\\ \hat{A}&...
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1answer
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Singular points of a matrix when the entries are restriced to a Lie Group

Let $\mathsf{SO}(3)$ be the set of $3 \times 3$ rotation matrices. Let $R\in\mathsf{SO}(3)$ and $r_{ij}$ represent the entry of $R$ sitting at the $i^{th}$ row and $j^{th}$ column, i.e., $$ R \in\ \...
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1answer
80 views

Do matrices $\mathbf{A A}^H$ and $\mathbf{A}^H \mathbf{A}$ have the same eigenvalues?

Let $\mathbf{A}$ be any complex matrix. Do matrices $\mathbf{A A}^H$ and $\mathbf{A}^H \mathbf{A}$ have the same eigenvalues? Note: The matrix $\mathbf{A}^H$ is the conjugate transpose of the matrix $...
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bounding $\|A^K x\|_2$ more tightly than $\|A\|^k_{op} \|x\|$ when $\rho(A) < 1$

Let $\|\cdot\|$ denote the Euclidean operator norm when applied to matrices, and the $L2$ norm when applied to vectors. Let $\rho(\cdot)$ denote the spectral radius (maximum magnitude of $A$'s ...
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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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30 views

Eigenvalue of product of a matrix with a non negative matrix.

Acording to this Now in my case, I have A=-LB, where L is a Laplacian matrix of an undirected graph. Hence L is a nonnegative matrix. Since the maximum singular value of (-L) is 0, can I conclude ...
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1answer
110 views

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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Singular Value Decomposition Basis?

I am unable to just understand one bit of SVD. Given A=USV where U is the eigenvectors of the correlation matrix between the entries S is the eigenvalue matrix and V the transpose of eigenvectors of ...
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1answer
31 views

Submultiplicavity of spectral norm

Is spectral norm (i.e. the maximum singular value of a matrix) submultiplicative? I am absolutely confused. How to express the singular value of the product of matrices in terms of that of the ...
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1answer
51 views

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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1answer
34 views

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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2answers
66 views

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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1answer
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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
40 views

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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1answer
86 views

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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1answer
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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
106 views

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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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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1answer
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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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1answer
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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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1answer
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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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2answers
63 views

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
34 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,...