Questions tagged [eigenvalues]
eigenvalues of matrices or operators
56 questions from the last 365 days
4
votes
1
answer
272
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
8
votes
1
answer
358
views
Eigenvalues of a certain combinatorially defined matrix
Let $A_n$ be the matrix whose rows and columns are indexed by pairs
$(i,j)$ with $1\leq i,j\leq n$ and $i\neq j$ (so $A$ is an
$n(n-1)\times n(n-1)$ matrix), whose $((i,j),(k,l))$-entry is 0 if
$i=k$ ...
- 50.8k
-5
votes
1
answer
86
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 23
-1
votes
0
answers
81
views
Can the higher order corrections to energy be calculated the same way in degenerate perturbation theory as with non-degenerate perturbation theory?
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Now from non-degenerate perturbation ...
- 23
0
votes
0
answers
46
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
4
votes
1
answer
90
views
Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products
Let $\mathbf{A}_i \in \mathbb{R}^{d \times d}$ ($i = 1, \dots, T$) be symmetric positive semidefinite (PSD) matrices. Define the quantity
$$
m = \frac{\lambda_{\max}\left(\sum_{i=1}^T \mathbf{A}_i^2\...
- 73
4
votes
1
answer
161
views
Finite group is Dedekind iff for every irrep, every element acts as identity or has all eigenvalues $\ne 1$
Consider the following claim: a finite group $G$ is Dedekind $\iff$ for every irrep $\rho$, and every $g \in G$, $\rho(g)$ either is identity matrix or has all eigenvalues $\ne 1$.
Is this claim true?
...
- 393
1
vote
3
answers
257
views
Eigenvalues of positive matrices in $\mathrm{SL}(d,\mathbb{Z})$
Let $A\in\operatorname{SL}(d,\mathbb{Z})$ be an irreducible positive matrix, i. e. $A=(a_{i,j})_{1\leq i,j\leq d}$ with $a_{i,j}\in\mathbb{Z}_{>0}$. From the Perron-Frobenius theorem, we know that $...
- 11
4
votes
1
answer
322
views
Minimum eigenvalue of a symmetric matrix
I was solving a problem and got stuck on the following:
Let $[p] = \{1, \ldots, p\}$ where $p \in \mathbb{N}$. Let $P(n, r)$ denote the set of all injective functions from $[r]$ to $[n]$ and write a ...
- 41
0
votes
0
answers
57
views
Class of covariance matrices invariant under permutations
I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:
\begin{equation}
U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^...
- 1
4
votes
1
answer
244
views
Distinct eigenvalues of random matrix over finite field
Let $A$ be a uniformly random matrix in $\mathrm{M}_n(\mathbf{F}_p)$.
It is well known that, as $p$ is fixed and $n$ tends to infinity, $A$ has repeated eigenvalues (over the algebraic closure $\...
- 309
4
votes
2
answers
587
views
Is this function injective?
For all given ordered lists $$\mathcal A=\big\{\{a_\mu\mid\mu=1,\cdots,N\}\mid\forall\mu,\nu> \mu,\ a_\mu > a_\nu\big\},$$ the function on the quotient space $$ G_\mu(a+\mathbb R: \mathcal A / \...
- 87
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
0
votes
0
answers
28
views
Constructing random graphs with given eigenvalues and eigenvectors
In Linial's presentation on SOME PROBLEMS AND RESULTS IN THE
GEOMETRY OF GRAPHS, on slide 7, some relations of properties of graphs to the eigenvalues of their adjacency matrix are listed, e.g.
if $G$...
- 13.2k
2
votes
0
answers
35
views
Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
0
votes
0
answers
15
views
Change in two spectral deviations due to edge deletion in a signed graph
Prove (or disprove) the following. Let $\Sigma=(G,\sigma)$ be a given signed graph. If $\lambda_1\ge\lambda_2\ge\cdots\ge \lambda_n$ and $\mu_1\ge\mu_2\ge\cdots \ge \mu_n$ are the eigenvalues of the ...
- 61
1
vote
0
answers
61
views
Eigenvalues and eigenvectors of the path Laplacian
Consider the Laplacian matrix of the path graph:
$$
L = \begin{bmatrix}
1 & -1 & 0 & \cdots & 0 & 0\\
-1 & 2 & -1 & \cdots & 0 & 0\\
0 & -1 & 2 & \...
- 11
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 91
7
votes
1
answer
274
views
Smallest eigenvalue of a random matrix
Let $A \in \mathbb R^{n\times n}$ be a positive semi-definite matrix,
and let $b \in \mathbb R^n$. For a random vector $x \sim \mathcal N(0, I_{n\times n})$, consider the random matrices
$$
B_1 = A + ...
- 211
2
votes
1
answer
172
views
Diagonalize almost symmetric tridiagonal matrix
I begin with an $n \times n$ real symmetric tridiagonal matrix. However, I replace the non-zero elements in the first and last rows with zeros, so it is no longer symmetric
$$M = \begin{bmatrix} 0 &...
- 151
2
votes
1
answer
754
views
On a combinatorial inequality
Is it true that
\begin{gather}
\min\left(\lambda_{\min}(M_{12}), \lambda_{\min}(M_{13}), \lambda_{\min}(M_{14}), \lambda_{\min}(M_{15}), \lambda_{\min}(M_{23}), \\ \lambda_{\min}(M_{24}), \lambda_{\...
- 178
4
votes
2
answers
283
views
Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
How accurate is the following statement: "For tree graphs, the multiplicity of the smallest non-zero eigenvalue $\lambda_2$ of the Laplacian is 1." If not valid, in which cases does it fail ...
- 91
4
votes
1
answer
103
views
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
- 145
4
votes
1
answer
199
views
First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
I am working with the matrix function
$$
f(A) = \frac{1}{\lambda_{\min}(A)},
$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
- 91
3
votes
1
answer
187
views
Eigenvalues of certain matrices
We write $R(\theta)=\left(\begin{smallmatrix}\cos(2\pi\theta)&\sin(2\pi\theta)\\ -\sin(2\pi\theta)&\cos(2\pi\theta)\end{smallmatrix}\right)$ for any $\theta\in\mathbb R$.
Let $d,m,n,r$ be a ...
- 2,446
1
vote
1
answer
54
views
Eigendecomposition of a Gram matrix with specified columns
I originally posted this question on Math SE, but I am starting to think it is more suitable to post it here. After waiting about two weeks, I didn't get any activity. The original question is linked ...
- 111
15
votes
1
answer
649
views
On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 178
7
votes
4
answers
557
views
Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 12.7k
0
votes
0
answers
22
views
Eigenvalues of Composition of Hadamard Operations of Low Rank Matrices
I am interested in the eigenvalues of $$ee^T \oslash (aa^T - a^{\odot2}(a^{\odot2})^T )^{\odot \frac{1}{2}},$$ where $a \in \mathbb{R}^n$ and $e$ is the vector with all entries equal to one.
Can we ...
2
votes
0
answers
101
views
What is the natural module?
Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...
- 217
12
votes
0
answers
301
views
Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?
Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
- 1,095
5
votes
2
answers
393
views
Maximal eigenvalue of a real symmetric Toeplitz matrix
The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$.
\begin{bmatrix}
n & n-1 & n-2 & \cdots & 1\\
n-1 & n & n-1 & \cdots & 2\\
n-2 & n-1 & n &...
- 61
0
votes
0
answers
77
views
Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero.
...
22
votes
0
answers
449
views
Why does the random shift in the QR eigenvalue algorithm work in the non-symmetric case over the complex field
I tried to implement the QR algorithm for non-symmetric matrices with complex entries to show to my students. The main part of the implementation was standard: the Householder reduction to the ...
- 61.9k
-1
votes
1
answer
62
views
An example of a matrix whose eigenvalues fullfill 'No-resonance' condition
No-resonance for a matrix is defined in terms of its eigenvalue as (last para page-3 in ref.):
$$\lambda_i \neq \sum_{j=1}^N m_j\lambda_j;\ \forall m_j\in \mathbb{Z}\ \ and\ m_j\geq 0$$
$$such\ that\ \...
- 101
0
votes
0
answers
92
views
Second largest eigenvalue of graph
Let $G$ be a connected $d$-regular n-vertex graph and let $k:= k(n)\in \mathbb{N}.$ Given a Non-empty set of vertices $\phi\neq B\subseteq V(G),$ how can I prove that all but at most $\frac{\lambda_2|...
- 53
0
votes
1
answer
131
views
Function of eigenvalues of Laplacian matrix
Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
- 53
3
votes
1
answer
87
views
Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real
Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...
- 1,295
1
vote
0
answers
64
views
Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
0
votes
0
answers
72
views
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 ...
- 297
0
votes
0
answers
32
views
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 ...
- 4,058
0
votes
2
answers
397
views
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 25
0
votes
0
answers
148
views
Is there a way to find the eigenvalues of a matrix using character table?
I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
- 1
1
vote
1
answer
48
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
2
votes
0
answers
81
views
Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
- 21
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
1
vote
0
answers
81
views
The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]
Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$.
It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
- 4,058
0
votes
1
answer
91
views
Matrix quantization and effect on singular values
Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for
$$
\|
\sigma_i(A)-\...
- 5,405
52
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 ...
- 12.7k