Questions tagged [eigenvalues]

eigenvalues of matrices or operators

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What are the convergence requirements for Inverse Power Method?

I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below. ...
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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] ...
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3 votes
2 answers
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Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration

I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
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When is the Fourier discrete matrix almost similar to some particular diagonal matrix?

Let $N$ be a natural number and put $z=e^{\frac{\pi i}{N}}$ and $w=z^2$. Let us consider the discrete Fourier matrix $F=(w^{kl})_{k,l=0,\cdots,N-1}$ and the diagonal matrix $D=\operatorname{diag}(1,...
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A matrix of character and trace

Let $q$ be a prime power. Let $g$ be a multiplicative generator of $F_{q^2}$, the finite field with $q^2$ elements. Assume that $l$ is a fairly large prime ($>q^4$) dividing $q^{2(q-1)}-1$. Let $\...
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The spectrum of Laplacian operator

Let $ \Omega $ be a bounded domain in $ \mathbb{R}^d $. For $ f\in L^2 $, it is well known that we have a unique solution $ u\in H_0^1(\Omega) $ by using Lax-Milgram theorem for the Dirichlet problem ...
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Find the eigenvectors from the QR algorithm in the unsymmetric case

It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$. I implemented a version ...
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Eigenvalues of Sturm–Liouville operator

Can we calculate the eigenvalues and eigenfunctions of the following operator in $W^{1,2}(\mathbb{R})$? $$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
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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 ...
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Proving 2 matrices have the same trace [closed]

I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is: Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...
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Minimum eigenvalue of normal matrix with polynomial basis

For each $n\in \mathbb{N}\cup\{0\}$, let $x_n(t)=\frac{t^n}{n!}$ for all $t\in [0,1]$. As the functions $X_N=(x_0 ,\ldots, x_N)$ are linearly independent, the matrix $ \int_0^1 X_N(t)^\top X_N(t)\,\...
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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 $!!$ ...
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Diagonalizing a symmetric block matrix

Let us consider the matrix $$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$ then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$ Now, let us consider a block matrix $$ A = \begin{...
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Maximal eigenvalue of a correlation matrix with some entries fixed as zeros

Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
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Kernel of the Laplacian + a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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Does Wilkinson's shift need to be discontinuous?

Given a symmetric Hessenberg matrix $A = \left[\begin{matrix}\ddots& \vdots & \vdots\\\dotsb & a & b\\\dotsb& b & c\end{matrix}\right]$, the Wilkinson shift $\mu$ employed in ...
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1 vote
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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-...
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6 votes
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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 $...
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4 votes
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Why do these polynomials split almost in the middle?

Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
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Best constant for Poincaré inequality on spheres

I am interested in the following Poincaré-type inequality, $$ \int_{S(r)} \lvert u-\bar{u}\rvert^2 d\sigma \leq C(N) \int_{S(r)} |u_{\theta}|^2 d\sigma$$ where $\bar{u} = \frac{1}{\lvert S(r)\rvert}\...
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3 votes
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Diagonalization of a specific Dirac operator

A few hours ago, a question was posed asking for the eigenvalues and eigenvectors of the Dirac operator $$ H=\begin{pmatrix} x & 0 & -i\partial_{x} & \bar{z} \\ 0 & x & z & i\...
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Relation between the dimension of vector spaces and dimension of the space [closed]

Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
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On an assertion in Payne's paper concerning the exterior Steklov eigenvalue problem

In the paper "New isoperimetric inequalities for eigenvalues and other physical quantities" by L. E. Payne [Communications on Pure and Applied Mathematics, 1956, 9(3): 531--542. https://doi....
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5 votes
1 answer
140 views

Log determinant of quadratic form

I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
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When must an eigenvector have only non-negative entries?

What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ...
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'Symmetric real matrix' analog in (Pseudo)Riemannian space

A symmetric real matrix has interesting properties, for example, its eigenvalues are real and eigenvectors corresponding to distinct eigenvalues, orthogonal. However, in (Pseudo)Riemannian space, if ...
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4 votes
4 answers
512 views

Under what conditions are the eigenvalues of a product of two real symmetric matrices real?

Under what conditions are the eigenvalues of a product $M = A B$ of two real symmetric matrices $A$ and $B$ real? And is there a way to relate the signs of the eigenvalues of $M$ to any properties of $...
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Asymptotics for the first eigenvalue for the Laplace-Beltrami operator on the sphere

I am trying to understand the existence of positive solutions for the following equations, $-\Delta_{\mathbb{S}^n} u + \lambda u = f(u)$ where $f$ is some non-linearity, say $f(t)=t^3.$ By considering ...
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What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?

What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$? I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
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Eigenvalues of large block matrices arising in quantum many-body problems

I have a real block matrix that looks like: $$ M = \begin{pmatrix} D_0 &C_{01} & 0 & ... & 0& 0 & 0 \\ C_{10} & D_1 & C_{12} & ......
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1 answer
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Change in the largest eigenvalue due to perturbation of diagonal components of a symmetric matrix

Let $A\in \mathbb{R^{n\times n}}$ be a symmetric negative difinite matrix and $D\in \mathbb{R}^{n\times n}$ be a diagonal matrix $D = \mathrm{diag}\{d_i\}, (d_i < 0)$. From Weyl's inequality, the ...
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4 votes
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151 views

Min max of a quadratic form of plus-minus ones

Does the following limit exist? $$ \lim_{n \to \infty}\, n^{-3/2} \min_{a_{ij}=\pm 1}\max_{x_{j}=\pm 1}\left|\sum_{1\leq i <j \leq n} a_{ij}x_{i}x_{j} \right| $$ There is no any significant ...
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2 votes
1 answer
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Limitation through the singular values

Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
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6 votes
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174 views

Typical eigenspectrum of a random projection of a large matrix

Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection $$ A =...
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Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
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1 answer
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Eigenvalues under linear transformation

Let $X$ and $Y$ be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small $\varepsilon>0$ such that, for any eigenvalue $\...
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eigenvalues of the product of a unitary with a diagonal

In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
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How to prove the Ky Fan inequality and its opposite

How do I prove that if $A$ and $B$ are Hermitian matrices with eigenvalues $a_1>a_2>\dots >a_n$ and $b_1>b_2>\dots >b_n$ and the eigenvalues of the sum are $c_1>c_2>\dots >...
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1 vote
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Perturbation theory for $UV^*$ in singular value decomposition

There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' ...
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1 vote
1 answer
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Verify that a given function is a fundamental solution to the heat equation on the hyperbolic plane [closed]

A few days ago (see this), I asked a question regarding the derivatives of the function $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$...
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1 vote
1 answer
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Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs

Consider the vector field $V:\mathbb{R}^4\rightarrow\mathbb{R}^4$, defined by \begin{equation} V(x,v,M_0,M_1)=(v,\kappa^{-1}(\beta M_0-v-kx),-M_0+v M_1,-M_1+1-vM_0), \end{equation} such that $\...
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2 answers
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Asymptotic for eigenvalues for the following ode?

Consider the following Sturm-Liouville problem, $$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$ where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$ I ...
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How does boundary perturbation affect the eigenvalues of differential equations?

There is a well-known procedure (at least to me) to compute how a small perturbation will affect the eigenvalues of a differential equation. However, the method deals only with perturbing the ...
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2 votes
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How to compute this limit involving the associated Legendre function?

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
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Largest eigenvalue of matrix A smaller than 1, what about B when A=B+C? [closed]

Suppose I have a square matrix $A$ that only has non-negative real entries and is not symmetric and not primitive either. It has no "special" structure we could exploit. I know that the ...
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4 votes
1 answer
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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 ...
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  • 407
12 votes
4 answers
635 views

Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix ...
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3 votes
1 answer
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Spectra of the Laplacian operator on the spherical space-form

Let $S^3/\Gamma$ be a spherical space form where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $S^3$. If $\Gamma$ is trivial, it is well-known that the spectra of the Laplacian operator on $...
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0 answers
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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 $Y$ ...
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Reference for eigenvalue error bound for "constrained" pencil

For a generalized eigenproblem (where $A$ is symmetric and $M$ is symmetric positive definite), we have \begin{equation} | \lambda - \theta | \leq \frac{ || A x - M x \theta ||_{M^{-1}} }{ || x ||_M } ...
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