All Questions
Tagged with eigenvector reference-request
10 questions
10
votes
2
answers
373
views
Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
- 311
7
votes
2
answers
5k
views
the complexity of Lanczos method
Hi, all
I am working on an algorithm which uses Lanczos method to compute K smallest eigenvalue(and their eigenvectos) of a sparse matrix, just want some information or links about the complexity of ...
- 73
7
votes
1
answer
464
views
"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...
- 884
4
votes
0
answers
137
views
What do we know about the generalized eigenvalue problem involving a projector?
Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
$$...
- 325
3
votes
1
answer
656
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,712
2
votes
2
answers
446
views
Entrywise modulus matrix and the largest eigenvector
Disclaimer. This is a cross-post from math.SE where I asked a variant of this question two days ago which has been positively received but not has not received any answers.
Let $A$ be a complex ...
- 623
2
votes
0
answers
43
views
References on discrete Sturm-Liouville eigenvectors convergence
Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
- 593
2
votes
0
answers
765
views
Reference: Continuity of Eigenvectors [closed]
I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer.
For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),...
- 1,724
2
votes
0
answers
339
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Sparse Eigenvectors for the Discrete Fourier Transform matrix
There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$.
Has there been any refereed work that ...
- 800
0
votes
0
answers
29
views
What is known about the distribution of eigenvectors for random positive semidefinite matrices?
Let $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$.
Define the random $d \times d$ real positive semidefinite matrix,
$$
S_n = \frac{1}{n} \sum_{i=1}^n ...
- 380