Questions tagged [singular-values]

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182 questions
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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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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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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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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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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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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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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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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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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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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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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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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 ...