Questions tagged [eigenvalues]

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39 views

Does the eigenvalue equality hold for my expression?

Let $g(\boldsymbol{\theta},\boldsymbol{\theta_0}) = trace [ \boldsymbol{\Omega{(\boldsymbol{\theta})}}^{-1} \boldsymbol{\Omega{(\boldsymbol{\theta_0})}}]-ln[det(\boldsymbol{\Omega{(\boldsymbol{\...
3
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1answer
34 views

Location of bulk and edges for Gaussian random matrices

I have some trouble to understand the difference between the "bulk" and the "edges" of the spectral density of random matrices (for instance in this question). From my understanding, all properties ...
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0answers
30 views

Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation \begin{equation} \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
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0answers
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+100

How to proceed in this Boundary value problem where Eigen values are calculated numerically?

While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$) \begin{eqnarray} \lambda_h F''' - 2 \...
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1answer
151 views

Do there exist graphs whose adjacency matrix is positive semi-definite? [closed]

If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
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54 views

Convergence to equilibrium of a nonlinear dynamical system

Consider the following dynamical system in $\mathbb{R}^n$ $$ \dot{x} = -x + A\tanh(x)=:f(x) $$ where $x = (x_1,...,x_n) \in \mathbb{R}^n$, $A$ is a real matrix with spectral radius $\rho(A) < 1$, ...
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0answers
16 views

Proof for strict separation of the eigenvalues ​of a Jacobian matrix with its minors

Let's consider a jacobi matrix (or tridiagonal symetric matrix where adjacent diagonals coefficients are strictly positive) : \begin{equation} T_n = \begin{bmatrix} a_1 & b_1 & 0 & \...
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0answers
40 views

Relationship between negative operator eigenvalues

Let $L>0$, $c \in (-1,1)$ and $\varphi \in H_{per}^{2}([0,L])$ be fixed. Define $w:= 1-c^2>0$. Consider the matrix operator $\mathcal{L}_{R}: H_{per}^{2}([0,L]) \times L_{per}^{2}([0,L]) \...
4
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2answers
236 views

Eigenvalues of a matrix sum

I have a control system problem, which ends up in that the eigenvalues of the system matrix should have a negative real part, then the system is stable. The system matrix is real but not symmetric. ...
4
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1answer
247 views

Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor

I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...
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0answers
29 views

Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements

Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric) $$ \begin{...
1
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1answer
42 views

How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results. Is there any method, which ...
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0answers
36 views

Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem: I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$. The first is a regular Toeplitz matrix $A$...
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0answers
20 views

Upper bound for eigenvalue of symmetric kernel

Let $V \in L^2(D \times D)$ be symmetric kernel defining the compact and nonnegative integral operator \begin{equation}\mathcal{V}: L^{2}(D) \rightarrow L^{2}(D), \quad(\mathcal{V} u)(x)=\int_{D} V\...
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6answers
1k views

Differentiability of eigenvalues of positive-definite symmetric matrices

Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
4
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1answer
109 views

Relation of row sums to largest eigenvalue

I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
1
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1answer
153 views

A linear algebra question regarding the eigenvalues of the product of a diagonal matrix and a projection matrix

I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few ...
2
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1answer
115 views

Eigenvalues and eigenvectors of Gaussian random matrices

Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues? ...
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0answers
50 views

$\lambda_2$ of Laplacian of a regular graph

Given a $d$-regular graph $G=(V,E)$ with $|V| =n$. We know that the smallest eigenvalue of the normalized laplacian matrix of $G$ is $0$. I have seen the formulation of the second smallest eigenvalue $...
1
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1answer
104 views

Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements

I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that $$ A = \begin{bmatrix} -\alpha & \alpha & -\gamma^{-1} & 0\\ \beta &...
1
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1answer
81 views

Eigenvalues of adjacency matrix of a k-regular graph

If $A_G$ is the adjacency matrix of a k-regular graph, let $B = J+xA_G$, where J is the matrix whose elements are all 1s and $x\in R$ is a scalar. If $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...
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0answers
28 views

Perturbation analysis and sensitivity of eigenvector matrix product with specific perturbation

In my research in applied linear algebra and probability (Wiener filtering) I have come across this rather interesting problem: For a matrix $ U $ we denote by $ U_k $ the matrix formed by taking ...
0
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1answer
59 views

Eigenvalues of an integral operator

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
2
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0answers
210 views

Characteristic polynomials of some special matrices

This is related to question Matrix-valued periodic Fibonacci polynomials. I want to find integer-valued matrices $x$ such that the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=...
2
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1answer
99 views

Common eigenvalues for two Sturm-Liouville problem

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form $$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
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0answers
41 views

Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
2
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0answers
95 views

Connections between eigenvalues of $B$ and $A+iB$

Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
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2answers
103 views

Differential equation satisfied by linear combinations of eigenfunctions of linear differential operator

Let $D$ be a linear differential operator on $\mathcal{C}^\infty(\mathbb{R})$, and let $\mathcal{E}_\lambda=\{f\in\mathcal{C}^\infty(\mathbb{R})|Df=\lambda f\}$ be the space of eigenfunctions of $D$ ...
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0answers
97 views

Link between eigenvalues of a symmetric matrix and a functional space

Let $f_1,\dots,f_n \in L^2(\mathbb{R},\mathbb{R})$ be $n$ mutually orthogonal functions with $\int f^2_i =1$ such that $|\{x \in \mathbb{R} | f_i(x) = 0\}| = 0$ for any $i \in \{1, \dots,n\}$. Does ...
4
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0answers
91 views

Inequality between Dirichlet and Neumann eigenvalue for Sturm Liouville problem

Consider the following Sturm Liouville problem on an interval $[a,b]$ $$\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{d} x}\right]+q(x) y=-\lambda w(x) y$$ for given ...
1
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2answers
105 views

Eigenvalues of tridiagonal symmetric matrix

Could you tell me please, are there any analytical methods how to find eigenvalues of matrix such this one? $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & 0 & \ldots & 0 \\ b_1 & ...
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0answers
34 views

Spectral abscissa of symmetric matrix with skew-symmetric perturbation

I am interested in bounds on the minimal distance between the spectral abscissa $\max_{\lambda\in\sigma(A)}\mathrm{Re}\lambda$ of a matrix $A$ and the eigenvalues of its perturbated version $A+S$. In ...
2
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2answers
147 views

Significance of the length of the Perron eigenvector

Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is ...
4
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0answers
139 views

Schrodinger operator with magnetic field: eigenvalues

Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, $$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$ where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
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0answers
85 views

What kernel function yields power law eigenfunctions

Suppose I have a kernel function $K(x, y)$. I can then define an integral transform as follows: $$K[f] = \int_0^\infty K(x, y) f(x) dx$$. Is there any kernel function where the eigenfunctions $f(x) =...
1
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1answer
146 views

Eigenvalue distribution of a band matrix

Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$. For some positive integer $k$, I define ...
6
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1answer
87 views

Maximum eigenvalue of a doubly stochastic matrix with deleted row and column

Consider an $n \times n$ irreducible and reversible (in the sense of a Markov chain) stochastic matrix $P$; assume that it has uniform stationary distribution (so, by reversibility, the matrix is ...
9
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0answers
706 views

Positive definiteness of matrix

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
9
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3answers
1k views

What happens to eigenvalues when edges are removed?

I am stuck at the following : Let $G$ be a graph and $A$ is its adjacency matrix. Let the eigenvalues of $A$ be $\lambda_1\le \lambda_2\leq \cdots \leq \lambda_n$. If we remove some edges from the ...
1
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1answer
139 views

Eigenvalues of product of symmetric positive definite matrices

Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$. Can I say $$ \frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n $$ If these matrices ...
2
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0answers
87 views

Relationship between eigenvectors of projected and original matrix

Let $A = \mathrm{Diag}(\lambda_1, \dots, \lambda_n)$ where $\lambda_1 \le \lambda_2 \dots \le \lambda_n$. Let $P = I - ww^T$ be a projection operator on an arbitrary $n$-dimensional hyperplane. Let $B ...
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0answers
183 views

Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it

In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
2
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2answers
72 views

Question about eigenvalues of connectivity matrices for graphs [closed]

I'm a computer science student working on a research project that deals with computational study of atomic clusters. I'm using a graph based representation of the clusters using a binary connectivity ...
2
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2answers
133 views

Is tridiagonal reduction the current best practice to compute eigenvalues of random matrices from the Gaussian ensembles (GOE, GUE, GSE)?

I have tried to compute the eigenvalues of random matrices of the GOE ensemble, using MATLAB. Such matrices of size $n * n $ can be obtained easily, symmetrizing matrices whose elements follow the ...
2
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0answers
67 views

Unimodality of a function of a non-negative matrix

I am taking an interest in the following problem: Consider a real matrix with non-negative entries $\boldsymbol{A} \in \mathbb{R}_+^{d \times d}$, with $d \in \mathbb{N}$. For $k \in \mathbb{N}$, ...
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3answers
211 views

When is it possible to find the sum of all elements of inverse of a matrix?

Given sum of elements of each row of a positive definite square matrix $M$ of order $n$ all of whose entries are non-negative, when is it possible to find the sum of all elements of the matrix $M^{-1}$...
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0answers
431 views

What are the eigenvalues of the sum of rank one matrices? [closed]

Consider the matrix $$ A = u_1 v_1^{\top} + u_2 v_2^{\top} + ... + u_n v_n^{\top} \in \mathbb{R}^{m \times m}, $$ where $u_i$, $v_i$ $\in \mathbb{R}^{m}$. Are there non-trivial conditions on $n$ ...
7
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2answers
355 views

Bounding the spectral gap of a simple symmetric matrix

I have a seemingly innocent linear algebra problem that I cannot solve, and which I hope that you would kindly offer some insight into. Here is the description: Let $\mathbf{a} = (a_1, a_2, \dots, a_d)...
5
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1answer
538 views

Eigenvectors of Kronecker Product [closed]

Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$. ...
3
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0answers
42 views

Spectrum of a symmetric saddle point matrix

Let $C=\left[ {\begin{array}{cc} A & B^{T} \\ B & O \\ \end{array} } \right]$, where $A\in \mathbb{R}^{n\times n}$ is SPD, $B\in \mathbb{R}^{m\times n}$ and $m\leq n$. The matrix $B$ ...

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