Questions tagged [eigenvalues]

eigenvalues of matrices or operators

Filter by
Sorted by
Tagged with
2 votes
0 answers
38 views

Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?

I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
BenJones's user avatar
2 votes
0 answers
125 views

Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
builtdifferential's user avatar
1 vote
0 answers
58 views

The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
ABB's user avatar
  • 3,898
0 votes
1 answer
67 views

Matrix quantization and effect on singular values

Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for $$ \| \sigma_i(A)-\...
ABIM's user avatar
  • 4,989
51 votes
8 answers
5k views

Is there a fast way to check if a matrix has any small eigenvalues?

I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues. I wish to extract from this list the tiny number of matrices that ...
Gordon Royle's user avatar
  • 12.3k
0 votes
0 answers
47 views

Induced higher Gershgorin estimate

I have a problem which I suspect appears in literature under a name I haven't found yet. Let $H:\ell^2(\mathbb{Z}^2)\to \ell^2(\mathbb{Z}^2)$ given by $H=\Delta + D$, where $\Delta$ is the graph ...
Keen-ameteur's user avatar
2 votes
1 answer
155 views

Sobolev regularity via Laplace spectrum

Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
ABIM's user avatar
  • 4,989
8 votes
2 answers
610 views

Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

It is known that an entire function that is nowhere zero must be the exponential of another entire function. Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
Kanghun Kim's user avatar
1 vote
0 answers
174 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
knuth's user avatar
  • 29
0 votes
1 answer
84 views

Change of the smallest positive eigenvalue after a rank-one update

Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$. Denote ...
Philipp Trunschke's user avatar
1 vote
0 answers
131 views

Eigenvalues of an Infinite Matrix - No Diagonal Dominance

I was wondering if anyone could help me or point me to resources to find the eigenvalue of the following infinite matrix: $g_{ij}=\text{exp}\left(\frac{-i j}{2}\right)$. Most resources I have found ...
Ellio's user avatar
  • 11
0 votes
0 answers
55 views

Chapter 2, Section 5 of Chavel's book “Eigenvalue In Riemann Geometry" is about the zero-point distribution of the derivatives of eigenfunctions

In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the ...
Ruifeng Chen's user avatar
2 votes
0 answers
167 views

A question about the regularity of the Schrödinger equation

While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem, \begin{cases} -\Delta u+Vu=\lambda u, &\text{in } \Omega \\ \...
Du Xin's user avatar
  • 31
4 votes
1 answer
82 views

Eigenvalues of the modified Mathieu equation with normalizable solution

The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ ...
renphysics's user avatar
0 votes
0 answers
32 views

Eigenvalues of minors to Schrodinger matrices

Suppose that we have a graph $G$, define the hamiltonian $H$ on it as $$Hu(x) = \sum_{y\sim x}u(y).$$ Consider the operator $H+V$ where $V$ multiplies the value $u(x)$ in any vertex by the potential ...
Станислав Крымский's user avatar
2 votes
0 answers
89 views

Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
Eduardo Longa's user avatar
0 votes
0 answers
47 views

Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...
SebastianP's user avatar
0 votes
0 answers
20 views

Nonnegative eigenvector form the point of view of variational characterization

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$ $(N\geq 3)$. We denote by $G(x, y)$ the Green function for the boundary value problem $$ -\Delta_x G(x, y)=c_n \delta(x-y) \quad \text { in } \Omega, ...
Davidi Cone's user avatar
1 vote
0 answers
80 views

Perron-Frobenius theorem to positive delay differential equations

The Perron-Frobenius theorem is that the largest eigenvalue (in modulus) of a non-negative matrix is real (and simple) and corresponds to a non-negative eigenvector. It is applicable to the positive ...
IscoBerlin's user avatar
1 vote
0 answers
125 views

Transforming nilpotency into diagonalizability [closed]

We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$. We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows: $Te_1=0$ and $...
ABB's user avatar
  • 3,898
6 votes
0 answers
165 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
Manuel Cañizares's user avatar
2 votes
1 answer
167 views

Eigenvectors and eigenvalues of a symmetric matrix and its entry-wise absolute value

The modulus of matrices is meant componentwise in the following. Let $H$ be a sqaure matrix that satisfies the following assumptions: $H$ is real-valued, symmetric, and positive-definite.. $H$ is ...
keisuke murota's user avatar
1 vote
0 answers
401 views

How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?

Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \begin{equation} \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
Aban's user avatar
  • 11
2 votes
1 answer
176 views

A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$

My question arises from Here. I have a series of eigenvalue equations in $B_R$. $$ -\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R, $$ where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\...
Elio Li's user avatar
  • 719
7 votes
3 answers
444 views

Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
Samuel Johnston's user avatar
4 votes
1 answer
175 views

Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D

Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
Eddie Lin's user avatar
  • 187
1 vote
1 answer
104 views

square matrix depending on complex value: spectral radius continous? [closed]

Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius. Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
Fynn13's user avatar
  • 83
2 votes
0 answers
216 views

Functional continuity of eigenvalues?

We have the following theorems! Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists ...
VSP's user avatar
  • 233
2 votes
1 answer
209 views

Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
Elio Li's user avatar
  • 719
1 vote
0 answers
115 views

Inequality concerning the imaginary parts of a recurrent sequence, Laplacian eigenvectors

Let $u=(u_1,\dots,x_n)\in\mathbb{C}^n$ be a sequence that satisfies the cyclic recurrence $$ \lambda+1 =a_{i-1}\frac {u_{i-1}}{u_i} + (1-a_{i+1})\frac{ u_{i+1} }{u_i } $$ with $a_i \in (0,1)$ and $\...
Artemy's user avatar
  • 650
1 vote
0 answers
77 views

Questions on the differential of the Lie logarithm

Let $G$ be a Lie group. Recall the Lie logarithm is well-defined about a neighborhood $U \subset G$ of the identity: $\log:U\to \mathfrak{g}$. I am dealing with a research problem that concerns the ...
Spencer Kraisler's user avatar
1 vote
1 answer
74 views

Infimum of the normalized Laplacian eigenvalues

Let $(M^n,g)$ be a compact Riemannian manifold. The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues: ...
Eduardo Longa's user avatar
6 votes
2 answers
245 views

Eigenvalues of polynomials of two matrices

In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting. Let $\chi(M)$ denote the characteristic ...
Brendan McKay's user avatar
0 votes
0 answers
83 views

Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix

Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is $$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\ ...
KAJ226's user avatar
  • 131
2 votes
0 answers
67 views

When is the reciprocal of an eigenfunction of the Laplacian on a domain $\Omega$ integrable?

Suppose that $\Omega \subseteq \mathbb{R}^n$ is a bounded domain and $u : \Omega \to \mathbb{C}$ solves $-\Delta u = \lambda u$ with Dirichet or Neumann boundary conditions. Can we say anything about ...
Dominic Shillingford's user avatar
0 votes
0 answers
83 views

Expectation of the operator norm of projection of a random permutation matrix

Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace. What bound can I ...
Kaiyue Wen's user avatar
1 vote
0 answers
60 views

Sub Laplacian on the quaternion Heisenberg group $\mathbb{H}$

The sublaplacian is defined by $\mathcal{L}=-\left(X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, which is independent of the choice of the orthonormal basis of $\mathbb{H}$. It is well known that ...
zoran  Vicovic's user avatar
0 votes
0 answers
63 views

Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant

Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse. ...
Uri Cohen's user avatar
  • 363
0 votes
0 answers
102 views

Can a laplacian-beltrami operator have negative eigenvalues?

Is it possible for an Laplace-Beltrami operator for Riemannian manifold to have negative eigenvalues? If not, are there any non-riemannian manifolds where one may observe negative eigenvalues for heat ...
learning_physics's user avatar
0 votes
0 answers
64 views

Eigenvalues of this Toeplitz matrix

I am looking for the analytic solution for the eigenvalues of a $(n+1)\times (n+1)$ matrix of the form $$ A_n=\begin{pmatrix} 1 & z & z^2 & z^3 & \cdots & z^n \\ z & 1 & z &...
Guoqing's user avatar
  • 431
0 votes
0 answers
64 views

Approximate solution problem of rank-one modification matrix secular equation

In Golub's paper , page 327,the eigenvalues of a rank-one modification of a $n\times n$ symmetric matrix can be computed by findng the zeros of the secular equation \begin{equation*} w(\lambda_j)=...
brant's user avatar
  • 43
10 votes
2 answers
445 views

Does approximate equality of quantum states imply operator inequality in a large subspace?

Let the trace norm of $X$ be $$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$ and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
Noel's user avatar
  • 165
1 vote
2 answers
275 views

Eigenvectors of a non-symmetric rank-one update of a symmetric matrix

I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...
brant's user avatar
  • 43
7 votes
0 answers
139 views

Subleading terms in Weyl's Law

The two term Weyl's conjecture states that $$N(\lambda)\sim\frac{\operatorname{area}(\Omega)}{4\pi}\lambda-\frac{\operatorname{perimeter}(\partial\Omega)}{4\pi}\sqrt\lambda$$ where $\Omega$ is a ...
antrep1234's user avatar
3 votes
0 answers
135 views

Eigenvalues of random matrices are measurable functions

I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable. If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
Curtis74's user avatar
0 votes
0 answers
71 views

When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?

By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that $$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$ where $e_1,\dots,e_n$ are the standard ...
ABB's user avatar
  • 3,898
2 votes
1 answer
122 views

Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?

By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that $$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that $$C=\...
ABB's user avatar
  • 3,898
1 vote
0 answers
134 views

Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by: $$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
Greyearl's user avatar
2 votes
1 answer
111 views

Maximizing the first Neumann eigenvalue on disks

Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that $$\mu_1(g) \operatorname{...
Eduardo Longa's user avatar
0 votes
0 answers
64 views

Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
  • 3,898

1
2 3 4 5
17