# Questions tagged [singularvalues]

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149 questions
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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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### Summation of singular values

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