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8
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0answers
445 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 ...
7
votes
0answers
155 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: ...
7
votes
0answers
120 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 ...
6
votes
0answers
396 views

Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...
5
votes
0answers
198 views

Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...
5
votes
0answers
156 views

Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem: ...
5
votes
0answers
103 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
5
votes
0answers
215 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
3
votes
0answers
51 views

Eigenspaces of the Laplace operator on a unit ball

I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal ...
3
votes
0answers
70 views

Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions: All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix); all principal minors are $>1$, ...
3
votes
0answers
101 views

Relating Numerical Range and Perron-frobenius theorem for positive matrices?

Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers $$W(A)=\{x^HAx\lvert ...
3
votes
0answers
109 views

Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...
3
votes
0answers
96 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 ...
3
votes
0answers
112 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 ...
3
votes
0answers
151 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 ...
2
votes
0answers
116 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
2
votes
0answers
35 views

Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications? For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature. In some of ...
2
votes
0answers
337 views

Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square ...
2
votes
0answers
113 views

Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?

Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.) I want to calculate or bound $\det ...
2
votes
0answers
109 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 ...
2
votes
0answers
204 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 ...
2
votes
0answers
90 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 ...
2
votes
0answers
174 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
0answers
281 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 ...
2
votes
0answers
209 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
0answers
754 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 ...
2
votes
0answers
89 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 $C$. ...
2
votes
0answers
231 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!) ...
1
vote
0answers
32 views

Characterization of eigenvector

Let's say we have the following optimization problem. (All the $\Sigma_{ii}$'s are positive definite.) $\max u^\top \Sigma_{12} v\quad$ $\text{subject to}\quad u^\top \Sigma_{11} u = 1\quad and\quad ...
1
vote
0answers
51 views

inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
1
vote
0answers
76 views

Eigenvalues of a partitioned self-adjoint matrix

This is a repost of the same question on MSE (with no reply/comment): http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix I would be grateful just for a ...
1
vote
0answers
163 views

3-regular (cubic) graph with adjacency eigenvalue 1

Suppose $A\in\{0,1\}^{n\times n}$ is the adjacency matrix of a 3-regular (cubic) graph $G=(V,E)$; that is, all $n$ vertices $v\in V$ in the graph have three neighbors. Is there a nice necessary ...
1
vote
0answers
69 views

Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph. Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
1
vote
0answers
97 views

Limit for eigenvalues of the Dirichlet problem

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem $$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, ...
1
vote
0answers
83 views

Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
1
vote
0answers
46 views

Bounds on the spherical measure of sub-level sets of quadratic forms

I'm wondering if there are any bounds on the spherical measure of sets of the form $$ \mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha) $$ where $\alpha$ is some ...
1
vote
0answers
78 views

Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy $$ \left\|A-Id\right\|_{op}<r $$ for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
1
vote
0answers
36 views

Eigenvalue overdetermined problem

Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$: $$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\ u = 0 ...
1
vote
0answers
139 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
1
vote
0answers
146 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
1
vote
0answers
82 views

An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...
1
vote
0answers
194 views

Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
1
vote
0answers
48 views

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
1
vote
0answers
240 views

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 ...
1
vote
0answers
124 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 ...
1
vote
0answers
105 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 ...
1
vote
0answers
224 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 ...
1
vote
0answers
127 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} \\\ ...
1
vote
0answers
63 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 ...
1
vote
0answers
89 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 ...