1
vote
0answers
30 views

Eigenvalues that are functions

Let us have the Laplacian on a compact manifold $M$. Suppose I have some equation of the form $$-\Delta u(x) = f(x)u(x).$$ If $f \equiv c$ were a constant, this would be an eigenvalue problem ...
5
votes
0answers
46 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
3
votes
2answers
49 views

“Sandwich theorem” for eigenvalues of symmetric matrices

I am looking for a reference for the following result for symmetric matrices Let $A\in\mathbb R^{n\times n}$ be symmetric with eigenvalues $\lambda_n \leq\ldots\leq\lambda_1,\, M\subset \lbrace ...
2
votes
0answers
59 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
9
votes
3answers
171 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
1
vote
1answer
43 views

What do real eigenvalues imply for a matrix

Suppose we have a matrix $A \in \mathbb{R}^{n \times n}$ with $\textrm{eig}(A)=\{ \lambda_1, \lambda_2, \ldots, \lambda_n\}$ such that $\lambda_i \in \mathbb{R}$. Does the realness of the eigenvalues ...
1
vote
0answers
102 views

Rank-2n tensor algebra eigenvalue equation

Im interested in resources and work done on the eigenvalue equation for rank-2n tensors: $$ M_{ij}A_{j} = \lambda A_{i} \\ $$ $$ M_{ijkl}A_{kl} = \lambda A_{ij} \\ $$ $$ M_{ijklmn}A_{lmn} = \lambda ...
1
vote
1answer
23 views

Layering of eigenvalues

If $\tau: V \to V$ is a linear transformation on a finite-dimensional real vector space with eigenvalues $a_1, \dotsc, a_n$, in ascending order, and $P:V\to V$ is orthogonal projection onto $W ...
5
votes
0answers
353 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
7
votes
0answers
167 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...
1
vote
0answers
193 views

Is there a great book on eigenvalues?

I keep encountering ostensibly very different branches of mathematics, only to have eigenvalues show up in each one. Is there a single book out there that presents a deep, unified account of the ...
10
votes
4answers
23k views

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...
2
votes
0answers
110 views

Second eigenvalue of a stochastic block matrix

Considering a stochastic block matrix in the form of, \begin{equation} \textbf{$P_{}$} = \left( {\begin{array}{cc} \textbf{$A_{}$} & \textbf{$B_{}$}~; \ \textbf{$B_{}$} & \textbf{$A_{}$} ...
3
votes
1answer
339 views

Eigenvalues of anti-circulant matrices using 1-circulant matrices

Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that, for any anti-circulant matrix, the ...
18
votes
2answers
16k views

A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter

A number of areas I'm studying in my degree (not a maths degree) involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the ...