Questions tagged [singular-values]

Question 1

I will write my question in the context of differential forms, but clearly this is problem can be stated much more generally. Consider the space of smooth differential $p$-forms on $\mathbb{R}^n$. ...

Question 2

Let $L$ be a general (possibly non-Hermitian) square matrix and define \begin{equation} A := e^{-i L}. \end{equation} I am interested in understanding how the singular values of $A$ relate to the ...

Question 3

Let $\sigma_i(M)$ denote the $i$'th singular values of matrix $M$, in decreasing order: $$\sigma_1(M) \ge \sigma_2(M) \ge \dots$$ Consider the matrix product $AMB$, where it is assumed that $A,M,B$ ...

Question 4

Let $M$ be a rectangular real matrix, and $A,B$ two diagonal real matrices with positive entries. I have two related questions: What are the singular values of $AMB$? Are they related in some way to ...

Question 5

Let $i$ be the imaginary unit and $A,B$ be two real $m\times n$ matrices. I wonder if there are any relations between singular values of $A+Bi$ and singular values of the real matrix $C:=\begin{...

Question 6

Problem: Find the value $x^*$ that minimizes the induced 2-norm $$ ||\Gamma - R(x)||_2, \; \; \Gamma \in \mathbb{R}^{2 \times 2}, \; \; R = \left( \begin{array}{cc} a + b x & -(b + ax) \\ b + ax &...

Question 7

I am investiganting some regularity properties for vector fields on $SU(2)$ and I reduced it to obtaining lower bounds on the absolute value of the eigenvalues (which coincide with the singular values)...

Question 8

I'm struggling to understand something about what I saw referred to as the lower norm function and its possible relation to singular values. Let $A\in \mathbb{R}_{m\times n}$ be a matrix with $m\geq n$...

Question 9

If anything, it will most likely decrease. Let $A$ be a matrix with $m$ rows and $n$ columns ($m\le n$) and $\sigma_m(A)>0$. Let $A'$ be the same matrix with one row removed. Their condition ...

Question 10

I've not used singular before, so I hope this question is not silly or trivial. I assume I have a finite nonempty real set $\mathbb{V}\subseteq \mathbb{R}$ and a potential function $V:\mathbb{Z}^2\to \...

Question 11

Let $x_1, x_2,\ldots,x_d$ be $d$ independent random vectors in $\mathbb{R}^d$ with entries sampled IID from the standard normal random variable $\mathcal{N}(0,1)$. Define the $d\times d$ random ...

Question 12

In the 4th edition of Linear Algebra Done Right by Sheldon Axler, the Singular Value Decomposition is stated and proved on page 287. With the way the proof is constructed, shouldn't it have been split ...

Question 13

Given $\mathbf{A} \in \mathbb{C}^{n\times m}$, let $\mathbf{A}^\dagger$ denotes its Hermitian transpose and $s_1(\mathbf{A}) = \underset{\|\mathbf{x}\|=1}{\mathsf{sup}} \|\mathbf{A}\mathbf{x}\|$ be ...

Question 14

I have a tall rank-$1$ matrix ${\bf Y} \in {\Bbb C}^{n \times m}$ (where $n > m$) whose singular value decomposition (SVD) is $$ {\bf Y} = \sigma_1 {\bf u} {\bf v}^\ast. $$ I add another $n \times ...

Question 15

What are the singular values $J_n(\lambda)$ Jordan block of size $n$ referred to the eigenvalue $\lambda$? I know there are a lot of estimates, but for a problem with only 2 parameters $n,\lambda$ ...

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Questions tagged [singular-values]

General formula for matrix representing the wedge product with respect to the lexicographical ordering

Singular values of $e^{-iL}$ in terms of properties of $L$

Majorization of singular values of $AMB$

Singular values of $AMB$, where $A,B$ are diagonal

How are singular values of $A+B i$ and singular values of $\begin{bmatrix} A & -B\\ B & A \end{bmatrix}$ related?

How to minimize a maximum singular value?

Eigenvalues of variation of the Kac matrix

Local minimizers of norm of a linear operator

Condition number of a matrix do not increase when data is removed

Relation of singular values of restriction to the spectrum

Why are these squared singular values distributed as $\exp(-\pi X^2)$ with Gaussian $X$?

Proof of singular value decomposition in Linear Algebra Done Right (LADR)

Bound on the maximal singular value of a complex matrix

Perturbation of singular values of rank-$1$ matrix

Exact singular values of a Jordan block

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