# 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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### 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 ...
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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### 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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### 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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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 $\... 6 votes 0 answers 174 views ### Typical eigenspectrum of a random projection of a large matrix Suppose I have a real symmetric$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 $$A =... 0 votes 0 answers 82 views ### Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let A be a real symmetry n × n matrix, with entries a_{ij}. For i∈{1,…,n} ... 1 vote 1 answer 113 views ### Eigenvalues under linear transformation Let X and Y be square non-symmetric matrices of the same size. Assume that their eigenvalues are close in the sense that there exists a small \varepsilon>0 such that, for any eigenvalue \... 0 votes 0 answers 80 views ### eigenvalues of the product of a unitary with a diagonal In M_n(\mathbb{C}), suppose U and D are a unitary and an invertible diagonal matrix with eigenvalues \{e^{i\theta_1},\cdots,e^{i\theta_n}\} and \{e^{i\eta_1},\cdots,e^{i\eta_n}\} ... 0 votes 0 answers 58 views ### How to prove the Ky Fan inequality and its opposite How do I prove that if A and B are Hermitian matrices with eigenvalues a_1>a_2>\dots >a_n and b_1>b_2>\dots >b_n and the eigenvalues of the sum are c_1>c_2>\dots >... 1 vote 0 answers 54 views ### Perturbation theory for UV^* in singular value decomposition There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' ... 1 vote 1 answer 77 views ### Verify that a given function is a fundamental solution to the heat equation on the hyperbolic plane [closed] A few days ago (see this), I asked a question regarding the derivatives of the function$$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)}}.$$... 1 vote 1 answer 164 views ### Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs Consider the vector field V:\mathbb{R}^4\rightarrow\mathbb{R}^4, defined by \begin{equation} V(x,v,M_0,M_1)=(v,\kappa^{-1}(\beta M_0-v-kx),-M_0+v M_1,-M_1+1-vM_0), \end{equation} such that \... 0 votes 2 answers 205 views ### Asymptotic for eigenvalues for the following ode? Consider the following Sturm-Liouville problem,$$(\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 ... 1 vote 0 answers 51 views ### 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 ... 2 votes 1 answer 61 views ### 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 ... 1 vote 0 answers 69 views ### 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 ... 4 votes 1 answer 222 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 ... 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 ... 3 votes 1 answer 132 views ### 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$...
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 $Y$ ...
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 } ...