All Questions
Tagged with eigenvalues sp.spectral-theory
82 questions
0
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50
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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 ...
- 633
1
vote
0
answers
183
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 ...
- 33
2
votes
0
answers
94
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 $\lambda_k(t)$, $...
- 2,095
6
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0
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201
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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 $\...
- 101
8
votes
0
answers
197
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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 ...
- 81
0
votes
0
answers
74
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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 $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
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0
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62
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What do you call this class of matrices with a unique positive eigenvalue associated to a graph?
I am looking for the name of a class of symmetric matrices $M\in\Bbb R^{n\times n}$ that I can associate to a (finite simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and that have the following ...
- 13.6k
4
votes
1
answer
167
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Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$
Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
- 481
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220
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Eigenvalue multiplicity of tensor product of positive operator with itself
Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
- 203
7
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195
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Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.6k
1
vote
1
answer
209
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Eigenvalues invariant under 90° rotation
Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \...
- 536
5
votes
1
answer
325
views
The discrete Fourier transform's Gaussian-like eigenvector
I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue?
$\begin{bmatrix}
2 & 1 & 0 & 0 & \cdots & 0 &...
- 1,547
5
votes
1
answer
260
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Resources on the stationary Schrödinger equation with the soliton potential
I am studying the following Lamé equation in the Jacobi form
\begin{equation}
-\frac{d^2 v}{dx^2} - \left(2k^2 \operatorname{cn}^{2}\left [x\;|\;k\right ]\right)v = \lambda v,
\end{equation}
...
- 51
2
votes
0
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539
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Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
2
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0
answers
121
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Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
16
votes
3
answers
2k
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Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 73
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0
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95
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Minimal eigenvalue of infinite dimensional matrix
Consider the following symmetric, positive-definite matrix
$$
H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}}
$$
where $n,m=0,1,2\dots$ (Here $!!$ ...
1
vote
1
answer
241
views
Monotonicity of eigenvalues II
In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
- 536
6
votes
1
answer
601
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Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
4
votes
1
answer
306
views
First eigenvalue of the Laplacian on the traceless-transverse 2-forms
Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.
Consider the first nonzero eigenvalue ...
- 891
0
votes
0
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109
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The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
1
vote
1
answer
195
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Eigenvalues of operator
In the question here
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend ...
- 192
3
votes
0
answers
284
views
Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
- 255
2
votes
0
answers
102
views
Spectrum of a Lax Pair and conservation laws of a PDE
I would like to ask a question that I had asked a few days ago on the site math.stackexchange
and I still have not received an answer.
If we have a Lax operator, we know that the spectrum of this ...
- 93
11
votes
1
answer
928
views
Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 124
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
0
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0
answers
124
views
Methods to find the spectrum of an operator
Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$?
Here is the setting I'm wondering about: consider ...
- 153
6
votes
0
answers
151
views
Gap between consecutive Dirichlet eigenvalues
Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say
$$ -\Delta \phi_k = ...
- 4,143
4
votes
0
answers
151
views
Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
- 651
6
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0
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107
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
4
votes
0
answers
98
views
Spectrum of Laplace-Beltrami with piecewise constant coefficients
By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
- 61
16
votes
2
answers
1k
views
Spectral symmetry of a certain structured matrix
I have a matrix
$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$
As ...
- 536
5
votes
2
answers
1k
views
Compact operator without eigenvalues?
Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$
$$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator!
Then, let $l$ be the left-shift and $r$ ...
- 173
12
votes
6
answers
2k
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 ...
- 3,423
1
vote
0
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67
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 $(...
- 4,547
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3
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273
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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 ...
- 175
4
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0
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155
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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\...
- 41
9
votes
0
answers
802
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 ...
- 192
2
votes
2
answers
264
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 ...
- 153
2
votes
0
answers
75
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 $k \in \mathbb{N}$, ...
- 141
1
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0
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52
views
Laplacian eigenvalue problem for systems coupled along the boundary
I am looking for references on eigenvalue problems for systems of the following type:
Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
2
votes
3
answers
999
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
0
votes
1
answer
262
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
user136156
3
votes
1
answer
463
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
- 536
2
votes
0
answers
150
views
Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
7
votes
1
answer
327
views
First eigenvalue of the spherical cap
Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ ...
1
vote
1
answer
77
views
Efficient way to compute eigenvalue decomposition for following problem
I have an optimization problem
$$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I
\end{array}$$
where $A\in R^{n \times n}$ and it is symmetric positive definite, ...
- 111
0
votes
0
answers
107
views
Numerical error on the spectrum of a matrix
Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
- 16.2k
1
vote
1
answer
177
views
Lower bound of the spectrum of a Schrodinger operator on a bounded domain
I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take ...
- 91
23
votes
1
answer
1k
views
Eigenvalues of Laplace operator
Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of ...
- 356