# Tagged Questions

The eigenvalues tag has no wiki summary.

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### 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 $\mathbf{S}$ $=$ ...

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220 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 ...

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**1**answer

108 views

### Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...

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301 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

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136 views

### Toda Flow Embeddings

What are strategies for generating the following types of pictures:
Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are:
...

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78 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 ...

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145 views

### Eigenvalues vs.matrix sparsity

For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity?
Is there a closed-form expression that states this ...

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86 views

### Eigenvalue problem

I am studying torsional Alfven waves in spicules.
In this concern I have encountered the following equation:
$
\left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m ...

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69 views

### What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...

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157 views

### Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...

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71 views

### Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...

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134 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 ...

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134 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 ...

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135 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 ...

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373 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 ...

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217 views

### Common eigenvector

I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem:
let $V$ be an infinite-dimensional locally convex (but not normed!) ...

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70 views

### Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...

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116 views

### Relationship betwen eigenvectors

Suppose that we have two matrices A and B. Matrix B is taken from A with one row and one column deleted. On the other hand A is n*n matrix and B is (n-1)*(n-1) matrix and is created by deleting last ...

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77 views

### Dominant eigenvalue of sum of tridiagonal and diagonal matrices

Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...

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111 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty ...

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65 views

### Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$).
Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$.
...

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96 views

### positiveness of the inverse solution to Sylvester equation

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\\
...

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60 views

### Which matrix/operator in a cone has the largest negative spectral part?

Background:
Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...

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74 views

### On generalization of Wigner semi circle

I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...

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111 views

### nodal lines in the dirichlet problem

In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help.

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76 views

### Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...

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65 views

### Eigenvectors of contraction times projection

Suppose $A$ is a real $n\times n$ matrix with real eigenvalues:
$$
1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0.
$$
Suppose $B$ is an involution, for simplicity let us assume that
$B$ is ...

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427 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} & & &\\
...

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114 views

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### Eigenvalue of a linear map over finite field

Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 ...

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29 views

### inverse of partial differential operator

I have a bounded degree hermitian partial differential operator over $\mathbb{R}^3$:
$D=\sum_{i,j,k,l,m,n\in{\{0,1,..5\}}} a_{i,j,k,l,m,n} x^iy^jz^k \frac{\partial^l}{\partial x^l} ...

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33 views

### Distribution of Wishart Sample Eigenvalues for Multiple Roots

I am interested in finding an asymptotic approximation to the latent roots $l_1>\dots>l_p$ of a white noise Wishart matrix $nS\sim W_p(n,I)$ as $n\rightarrow\infty$ (where $p$ is fixed). In ...

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### Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...

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### Eigenvalues of a “Half-Kronecker ” Product

The Problem:
Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other
2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given).
Now we can construct a new matrix ...

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### 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 ...

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### 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 ...

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### Laplacian using SDP

Is there any suggestion about how could one construct a model that uses semidefinite programming that minimizes sum of k smallest eigenvalues of Laplacian matrix?
I found two papers that have done ...

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236 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?

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90 views

### Ratio of Eigen values and Mutual Independence

Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix.
If $X$ contains $M$ observations ...

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239 views

### Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...