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2
votes
1answer
83 views

Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...
1
vote
1answer
64 views

Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...
5
votes
1answer
179 views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
0
votes
1answer
29 views

Using Marchenko - Pastur type Theorems on Regression Analysis

Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g_{p_n}(x) = \sum_{k=1}^{p_n} ...
5
votes
2answers
191 views

Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = ...
2
votes
0answers
113 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I want ...
2
votes
2answers
176 views

Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
2
votes
3answers
228 views

LU decomposition

Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...
0
votes
0answers
88 views

How could I prove this equality for eigenvalues of Laplacian matrix?

I would be glad if you have some comments that how I could prove following statement. Suppose that graph $G =(N, E)$ be given. The the following program computes the $k$-smallest eigenvalues of the ...
4
votes
1answer
211 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
3
votes
0answers
74 views

Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
2
votes
0answers
112 views

Distributions of eigenvalues for matrix normal distribution: related references

I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution. I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...
1
vote
2answers
97 views

Is Rellich's function valued theorem valid for a rank defficient function valued matrix?

Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends on $t$ analytically. (i) The $n$ roots of the characteristic ...
7
votes
1answer
694 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 ...
0
votes
0answers
47 views

Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...
5
votes
1answer
209 views

Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each ...
2
votes
0answers
116 views

What would be a better method for numerical diagonalization of a certain Vandermonde-like matrix?

For the fractional iteration of the $\exp()$-function Hellmuth Kneser had 1942 proposed an analytic solution valid on the real numbers; there is a technical implementation for Pari/GP of this method ...
2
votes
1answer
148 views

When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
2
votes
0answers
258 views

eigenvalues of a symmetric tridiagonal matrix with zero diagonals

I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements): $$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a ...
4
votes
1answer
227 views

A complex sequence with positive values

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d ...
2
votes
2answers
149 views

Is my use of the eigendecomposition correct here?

I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
4
votes
1answer
191 views

Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that $$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...
-1
votes
1answer
199 views

Upper bound on iterations count for power iteration algorithm

I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...
3
votes
1answer
204 views

Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question. Let $M$ be a non-symmetric real matrix: for example, the shear matrix $\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$. There ...
0
votes
2answers
220 views

Eigenvalues of an amplification matrix

Let $A$ and $B$ square real matrices. I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1. Can we say something about the eigenvalues ...
4
votes
2answers
177 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
2
votes
1answer
474 views

Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
0
votes
0answers
153 views

Summation of eigenvalues of tri-diagonal matrix smaller than specific value

Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general ...
0
votes
1answer
921 views

Can the first non-zero eigenvalue of a Laplacian matrix with more than 1 zero valued eigenvalue be used to reorder an adjacency matrix?

I have a graph with multiple connected components, and its adjacency matrix. I form the Laplacian matrix (wiki Laplacian matrix), and from the 1K nodes there around ...
0
votes
0answers
193 views

Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative? Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
0
votes
1answer
389 views

eigenvalues of a diagonal matrix times a matrix

Suppose we are multiplying matrix $A$ with a diagonal matrix $D$ from left, i.e., $X=D A$ where $D$ is a diagonal matrix with elements $$d_{ii}=\frac{1}{2} \text{ for } i=1,n$$ ...
3
votes
1answer
560 views

Connection between eigenvalues of matrix and its Laplacian.

Hello! There are two definitions of graph spectrum: 1) Eigenvalues of adjacency matrix $A$. 2) Eigenvalues of Laplacian of adjacency matrix ($L$). Different sources offer different properties based ...
1
vote
0answers
359 views

Eigenvalue problem for symmetric block tridiagonal matrices?

Is there a procedure to find the eigenvalues of $\textbf{M}$? ‎ $$\begin{eqnarray} ‎\textbf{M}=\left[‎ ‎\begin {array}{ccccc}‎ ‎\textbf{A} & \textbf{B} & & &\\‎ ...