Questions tagged [eigenvalues]
eigenvalues of matrices or operators
900 questions
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
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Problem about the variational formula for the principal eigenvalue of non self-adjoint 2nd linear elliptical operator
Let $P$ be a second order elliptic operator defined in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n, n \geq 2$. $\lambda_0$ is the principal eigenvalue of the Dirichlet eigenvalue problem
$$
...
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1
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How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
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2
votes
1
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Eigenvalues of a sum of tensor product of representation matrices
Let $G$ be a finite group and $H\le G$. Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of inequivalent, irreducible representations of $G$, where $\rho_1$ is the trivial representation ...
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Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that
$$
\underset{P^2 = P,\; \text{rank}(P) = p}{...
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Characterization of positive definiteness of product of two positive definite matrices with special structures
Let $A$ be $n\times n$ matrix with special structure $A:=I_n+a\cdot \mathbf{1}\mathbf{1}^{\rm T}$, where $I_n$ is $n\times n$ identity matrix, $a>0$ is a scalar and $\mathbf{1}$ is an $n$-...
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Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem stems from a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the similarity transforming matrix from ${\bf J}$ to ${\bf ...
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7
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2
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467
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A prototypical problem for transfer matrix calculations in combinatorics
Let $G$ be a directed graph with self-loops on the positive integers: for every $n$, $G$ has the directed edges $(n,n)$ and $(n, n+1)$; additionally, if $n$ is even, $G$ has the directed edge $(n, n/2)...
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Does the Bochner Laplacian have discrete spectrum when the curvature is non–degenerate?
On a compact manifold the Bochner Laplacian has a discrete spectrum. I'm wondering under what conditions this extends to non–compact manifolds. If the connection is the zero one-form on the trivial ...
- 1,187
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245
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Independence of parameter for eigenvalues of periodic family of tridiagonal matrices
Consider the family of matrices $C(\ell,\theta)\in \mathbb{R}^{2\ell+2\times2\ell+2}$, where $\ell\in\mathbb{N}$ and $\theta\in\mathbb{R}$ given by
\begin{equation*}
C(\ell,\theta)=\begin{pmatrix}
...
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1
vote
1
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264
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Exact form of eigenvalues of pentadiagonal Toeplitz matrices
The tridiagonal Toeplitz matrices
$$\begin{pmatrix}
a & b & & \\
c & \ddots & \ddots \\
& \ddots & \ddots & b \\
& & c ...
0
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0
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144
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Can it be proved that the specific real symmetric matrix is positive definite? (Numerically confirmed)
${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as
$$\...
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Numerical tests show the smallest eigenvalue of a certain matrix remains invariant when some parameters vary – any proof ideas?
I've numerically verified that the smallest eigenvalue of a specific matrix remains invariant when some parameters change. Looking for theoretical proof approaches. Appreciate any insights.
${\bf A} \...
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Inverse of a fourth order tensor and derivative of a tensor with respect to another using spectral decomposition
I am writing a UMAT code in Abaqus to simulate hyperelastic fracture. Towards that end, I have calculated the Second Piola-Kirchhoff tensor and the material consistent jacobian tensor. However, Abaqus ...
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Is the maximal eigenvalue of this $n$ by $n$ matrix $\left\lfloor \log _2(n)\right\rfloor$?
The Mathematica 14 program for computing the matrix $T$ is
...
- 1,203
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Existence of solution(s) for this Steklov-Neumann eigenvalue problem
Let $\Omega \subset \mathbb{R}^2$ be a bounded simply connected domain with piecewise smooth boundary $\partial \Omega = \Gamma \cup \Sigma$, where $\Gamma$ and $\Sigma$ are smooth curves. Let $F \in ...
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113
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Is it possible to analyze the eigenvalue of a specific tridiagonal matrix?
I'm considering the $n \times n$ tridiagonal matrix
$$ A = \begin{pmatrix}
0 & 1 & & & \\
1 & c & 1 & & \\
...
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1
vote
1
answer
165
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How smoothly does the first eigenvalue vary w.r.t. a variation of domain?
Let $\Omega\subset \mathbb R^n$ be a bounded open set with Lipschitz boundary, and for each $t\in[0,1]$, let $\phi_t:\Omega\to \mathbb R^n$ be an embedding (i.e. $\phi_t:\Omega\to \phi_t(\Omega)$ is a ...
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88
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Deformation of Gram matrix: determinant and eigenvalues
I want to compare the determinant and the eigenvalues of two gram matrices obtaining by deforming the first one by positive definite matrices.
Let us consider a family $(x_i)_{1\leq i \leq m}$ of ...
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145
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Why does adding a normalizing condition to an eigensystem not affect the eigenvalues?
I have been studying eigensystems in the context of boundary value problems, and I encountered a situation where a normalizing condition, like Eq. (2.17) in Derek E. Moulton,
Paul Grandgeorge and ...
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153
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Slight skew in the distribution of eigenvalues for normal symmetric matrices
I've just started tinkering with random matrix theory, and to do so I've been performing some simulations in R.
Something I've noticed is that the distribution of eigenvalues shows slight skew from ...
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190
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Bauer-Fike theorem
I have a doubt about the interpretation of the Bauer-Fike theorem. It states that:
Given $ A \in \mathbb{C}^{N \times N} $ diagonalizable matrix ($ A = S D S^{−1} $ and $ D $ diagonal matrix having ...
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81
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Unbiased estimator of the singular values of a large matrix
Suppose I have a large matrix ${\bf X} \in {\Bbb R}^{m \times n}$. By independently sampling $r$ rows and $c$ columns of $\bf X$, we obtain a random submatrix $\bf S$. From $\bf S$, how to obtain an ...
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87
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Pairwise comparison of the eigenvalues of two trace-class bounded self-adjoint operators
Assume that $C$ is a positive trace-class bounded self-adjoint operator over a real separable Hilbert space $H$. For $Q$ a positive self-adjoint bounded linear operator, with norm smaller than $1$, ...
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76
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Existence of a perturbation preserving positivity and spectral bounds for a positive linear map on symmetric matrices
Let $ T: \operatorname{Sym}^{d \times d} \to \operatorname{Sym}^{d \times d} $ be a linear map that is positive, meaning that if $ \mathbf{X} \in \operatorname{Sym}^{d \times d} $ is positive ...
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Karhunen–Loève expansion of Wiener process on arbitrary indexing interval $[a,b]$
A quick look at the Wikipedia page for Kosambi–Karhunen–Loève theorem shows how to compute the following expansion for the zero-mean Wiener process $(W_t)_{t\in[0,1]}$ with covariance kernel $K(s,t)=s\...
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Eigenvector quadratic form inequality
Given matrix $B\in\mathbb{R}_+^{n\times n}$ and scalar $\alpha \in \mathbb{R}_{+}$, let $A:=\alpha B+B^T/\alpha$. Note that $B$ and $A$ have nonnegative entries and that $\alpha$ controls degree of ...
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Eigenvalues of block matrix with $A$ on diagonal blocks and $B$ on off-diagonal blocks [duplicate]
Let $A, B \in {\Bbb R}^{2 \times 2}$ be two self-adjoint matrices. I am interested in the following block matrix
$$ M =
\begin{bmatrix}
A & B & B & \dots & B \\
B & A & B & ...
9
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1
answer
457
views
Localization of eigenvalues on complex plane
Let $B$ be a cyclic upper-triangular nonnegative matrix,
$$B = \begin{bmatrix} 0 & b_1 & 0& \dots &\dots &0 \\
0 & 0 & b_2 & 0& \dots & 0\\
\vdots &\vdots&...
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Dimension of the first eigen space of the Laplacian
This question might sound a bit vague, nevertheless I hope this still attracts the right crowd. Estimating the first eigen value of the Laplacian in any context is a very important question and there ...
- 1,146
2
votes
1
answer
168
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Variance of largest eigenvalue of Bernoulli matrix
We say that an $n\times n$ matrix $A_n$ is a random symmetric Bernoulli matrix if each entry $a_{ij}$ is $\pm 1$ with probability $1/2$, the entries $a_{ij}$ with $i\ge j$ are independent and $a_{ij}=...
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Determining conditions of Equilibrium point after perturbations
Let $ A $ be an $ n \times n $ matrix. The stability of an equilibrium point is determined by the eigenvalues of the Jacobian matrix $ A $. More specifically, if all eigenvalues have negativ real ...
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1
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436
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Show $2\times 2$ matrix has positive eigenvalues
Let the real matrix $\bf A$ be positive definite and let $c>0$. Consider the matrix
$${\bf M} := \begin{bmatrix}
0 & -\text{A}_{11}-\frac{c^2}{4} & \frac{c}{2} & -\text{A}_{12} \\
1 &...
1
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1
answer
131
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Diagonalizing a linear self-adjoint matrix in a rank-deficient non-standard inner product space
I want to solve $Ax=\lambda x$ in a rank-deficient non-standard inner product space $\mathcal{S}^r$.
By "rank-deficient", I mean that although $x$ is represented by a vector of $\mathbb{R}^n$...
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Bounds on the eigenvalues of perturbations of a symmetric matrix
This question was previously posted on MSE.
Let us fix $\varepsilon\in (0,1)$ and $\beta\in\mathbb R$.
Consider the $2 n\times 2n$ symmetric tridiagonal probability matrix
$$Q_n :=\begin{bmatrix}
1-\...
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votes
1
answer
99
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Two SPD matrices are identical?
Two SPD matrices admits eigen-decomposition $\Sigma_p=U_p S_p U_p^{\top}$ and $\Sigma_q=U_q S_q U_q^{\top}$, where $S_p$ and $S_q$ contain ordered eigenvalues that are distinct. Let $\Sigma_v=U_q^{\...
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Let $A$ be a SPD matrix. Suppose diagonal $(A)_{ii}$ equals to its eigenvalue $\lambda_i$. Must $A$ be a diagonal matrix?
Given a symmetric positive definite (SPD) matrix with eigendecomposition $A = U \Lambda U^{\top}$ follows $\operatorname{diag}(A)=\operatorname{diag}(\Lambda)$, is it necessary to have $U=I$ so that $...
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On computing the condition number of SPD matrices via convex optimization
Suppose that we have an $n \times n$ symmetric positive definite (SPD) matrix $\bf Q$ and that we would like to compute its condition number via convex optimization. In section 3.2 of Boyd et al.$^\...
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250
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eigenvalues spectrum of random matrix expressions using determinant identities
Given a random matrix $X$ (e.g., with i.i.d. Gaussian entries) and two matrix expressions $A(X)$ and $B_\lambda(X)=B(X,\lambda)$ which satisfy (for any instance of X):
$$0=\det(\lambda I-A(X)) \iff 0=\...
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How to justify $\mathbf{S}_{x}= \frac{E_{s}}{N}I_{N\times N}$ as the best choice when the channel matrix is unknown?
I am a graduate student working in Wireless Communication, studying random matrix theory and its applications. In the context of determining channel capacity, I encountered the following generalized ...
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votes
3
answers
423
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Symplectic inner products of eigenvectors of complex symplectic matrix
Let $S$ be a $2N\times 2N$ complex symplectic matrix ($\operatorname{Sp}(2N,{C})$) satisfying $S^\dagger JS=J$, where
$$J=\begin{pmatrix}0&I_N\\-I_N&0\end{pmatrix}.$$
The symplectic inner ...
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0
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74
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Mapping the magnitude of a complex quadratic form to a Hermitian quadratic form
I have a function $f(X) = |a^T X^{-1} a|$ that maps a complex symmetric matrix $X = X^T \neq X^H$ to a real number. I would like to perform optimization involving this function, and I try to convert $...
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0
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131
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Minimum eigenvalue and semidefinite cone
I have an linear matrix inequality(LMI) in the form: $G + x_1F_1 + \cdots + x_nF_n \succeq 0$, where $G$ and $F_i$ are symmetric matrices, $x_i \in \{0, 1\}$, and a matrix $A \succeq 0$ means that the ...
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2
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653
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Lowest eigenvalue of Toeplitz matrices: strategies?
Let $\{a_n\}_{n\in \mathbb{Z}}$, $a_n\in \mathbb{R}$, be such that $a_n = O(1/n^2)$ and $a_{-n}=a_n$. The Toeplitz matrix $A_N$ is the $N$-by-$N$ matrix defined by
$$A_{N,i,j} = a_{|i-j|}$$
for $1\leq ...
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4
votes
1
answer
409
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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:\...
- 22k
8
votes
1
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440
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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
$i=k$ ...
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-5
votes
1
answer
152
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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 ...
- 45
0
votes
0
answers
77
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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 \...
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4
votes
1
answer
152
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\...
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4
votes
1
answer
250
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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?
...
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