# Questions tagged [eigenvalues]

The eigenvalues tag has no usage guidance.

520
questions

**-1**

votes

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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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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; \...

**1**

vote

**0**answers

48 views

+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 \...

**1**

vote

**1**answer

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

**2**

votes

**0**answers

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 $\rho(A) < 1$, ...

**0**

votes

**0**answers

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 & \...

**-1**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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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**0**answers

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**

vote

**1**answer

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 ...

**1**

vote

**0**answers

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 $A$...

**0**

votes

**0**answers

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\...

**10**

votes

**6**answers

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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?
...

**0**

votes

**0**answers

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**

vote

**1**answer

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**

vote

**1**answer

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 $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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)-\...

**1**

vote

**0**answers

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**

votes

**0**answers

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,...

**1**

vote

**2**answers

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 $D$ ...

**0**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**2**answers

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 & ...

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**0**answers

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\...

**0**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**3**answers

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**

vote

**1**answer

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**

votes

**0**answers

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 ...

**11**

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**0**answers

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**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**0**answers

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 $k \in \mathbb{N}$, ...

**-2**

votes

**3**answers

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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vote

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 $B$ ...