Questions tagged [singular-values]

This tag is for questions relating to 'Singular Value'. The term “singular value” relates to the distance between a matrix and the set of singular matrices

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For non compact operators, is countability of singular values equivalent to countability of eigenvalues?

Generally speaking, compact operators have countable eigenvalues and countable s-values (or singular values). What about the reverse? If I know that a (non-compact) operator has countable singular ...
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On what condition on $L$, there exists unit vector $x$, such that $\Re(x^∗Lx)<0$ and $\Im(x^∗Lx)=0$?

Given $L\in\mathbb{C}^{n\times n}$. Question 1: On what condition on $L$, there exists unit vector $x\in\mathbb{C}^n$, such that $x^*L^*Lx=1$. Question 2: On what condition on $L$, there exists unit ...
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On the characterization of the least singular value

Let $A$ be an $m\times n$ real or complex matrix. Here it is shown that $$\sigma_{min}(A)=\min_{x\in F^n, \|x\|=1}\|Ax\| \quad \quad (1)$$ where $\sigma_{min}(A)$ denotes the "least" ...
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Expected norm of matrix product with random unit vector

Let $S_d = \{x \in \mathbb{R}^d : \|x\| = 1\}$ be the unit sphere in $\mathbb{R}^d$. Let $A \in \mathbb{R}^{n \times d}$. Note that \begin{align} \max_{x \in S_d} \|A x\| &= \max \sigma(A) &...
Motivation: I have two different matrices in $\mathbb{R}^{1000 \times 2048}$. $A_1$ is coming from an sparse optimization process whose objective is creating as much as zeros in $A_1$. In this sense, ...