All Questions
7 questions
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222
21
votes
4
answers
9k
views
Condition for two matrices to share at least one eigenvector?
Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $...
- 403
10
votes
2
answers
4k
views
Perturbation theory for the generalized eigenvalue problem
Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)...
- 1,193
9
votes
1
answer
3k
views
Frobenius-Perron eigenvalue and eigenvector of sum of two matrices
Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
- 403
4
votes
1
answer
1k
views
Why are 1 and -1 eigenvalues of this matrix?
This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...
- 403
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
3
votes
3
answers
358
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
