# Questions tagged [eigenvalues]

eigenvalues of matrices or operators

824
questions

3
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### Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real

Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...

0
votes

0
answers

32
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### Derivatives of a rotation tensor intended to rotate bases of a symmetric 2nd order tensor [closed]

Let $A = \sum_i \lambda_i N_i \otimes N_i$ and $B= \sum_j \mu_j n_j \otimes n_j$ , be the two symmetric and independent 2nd order tensors and their respective spectral decompositions. Let R be the ...

1
vote

0
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57
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### Reference request for non-banded Toeplitz matrix

I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...

2
votes

0
answers

74
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### Smallest eigenvalue of certain PD matrix decreases under sparse perturbation

Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...

0
votes

0
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### Initial guess in shifted QR algorithm

I'm comparing timings of two implementations of algorithms for the computation of Gauss-Legendre nodes.
1 - The first is a Newton algorithm to find the roots of the Gauss-Legendre polynomials. Quite ...

0
votes

0
answers

31
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### Eliminating nullity for enhanced non-singularity

If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...

0
votes

2
answers

352
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### How to show the following matrix has eigenvalues $-d,-d+1,...,d$?

Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...

0
votes

0
answers

110
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### Is there a way to find the eigenvalues of a matrix using character table?

I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...

1
vote

1
answer

39
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### Iteration matrix representation with complex conjugate operator

I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...

2
votes

0
answers

67
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### 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 ...

2
votes

0
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129
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### 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 $\...

1
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0
answers

64
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### 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 ...

0
votes

1
answer

72
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### 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)-\...

51
votes

8
answers

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### 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 ...

0
votes

0
answers

47
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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 ...

2
votes

1
answer

177
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### 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 ...

8
votes

2
answers

634
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### 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 ...

1
vote

0
answers

176
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### 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 ...

0
votes

1
answer

90
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### 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 ...

1
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0
answers

137
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### 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 ...

0
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0
answers

56
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### 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 ...

2
votes

0
answers

176
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### 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 \\
\...

4
votes

1
answer

104
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### 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 $\pm iz$:
$$y''-(a-2q\cosh(2z))y=0.$$
...

0
votes

0
answers

35
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### 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 ...

2
votes

0
answers

89
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### 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 $\lambda_k(t)$, $...

0
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0
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49
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### 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-...

0
votes

0
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21
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### 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, ...

1
vote

0
answers

82
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### 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 ...

1
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0
answers

125
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### 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 $...

6
votes

0
answers

175
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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 $\...

2
votes

1
answer

173
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### 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 ...

1
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0
answers

470
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### 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}}...

2
votes

1
answer

177
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### 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 $\...

7
votes

3
answers

466
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 ...

4
votes

1
answer

187
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### 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 $\lambda_i \ne 0$ ...

1
vote

1
answer

110
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### 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 ...

2
votes

0
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277
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### 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 ...

2
votes

1
answer

215
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### 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 ...

1
vote

0
answers

117
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### 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 $\...

1
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0
answers

84
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### 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 ...

1
vote

1
answer

76
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:
...

6
votes

2
answers

249
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### 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 ...

0
votes

0
answers

88
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 \\
...

2
votes

0
answers

70
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 ...

0
votes

0
answers

97
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 ...

1
vote

0
answers

62
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 ...

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.
...

0
votes

0
answers

104
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 ...

0
votes

0
answers

87
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 &...

0
votes

0
answers

67
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### 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)=...