# Questions tagged [eigensystem]

An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

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### Eigenvectors of Laplacian

I am studying introduction to Multigrid methods. In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as $w_k(x_i) = \sin(k \pi x_i),$ where $x_i$ is a ...
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### Oscillating eigenvectors for 2d-laplace operator

I am trying to calculate the eigenvectors of a Laplace-operator in 2d, with boundary conditions equal to $u=0$ if $x, y$ are outside of a rectangle defined as $(0.5, 0.5), (1.5, 1.5)$. For that I used ...
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### LOBPCG bad preconditioned performance for largest eigenpairs

The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem $$Ax = \lambda B x$$ where $B$ is symmetric and positive-definite, $A$ is symmetric. One of the features that makes LOBPCG so ...
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### Eigen-decomposition one eigenpair by one eigenpair?

Is it possible to conduct an Eigen-decomposition of a matrix one eigenpair by one eigenpair? And related to this question, what is the time complexity of truncated eigendecomposition? I am trying (...
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### Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation?

I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ...
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### Solving a time-independent 2D Schrödinger equation using PDE Toolbox in MATLAB

I want to solve the following PDE eigenvalue problem, $$-\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\psi_n(x,y)+x^2y^2\psi_n(x,y)=E_n\psi_n(x,y)$$ inside a square with ...
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### Generalization of eigendecomposition problem

Let $A\in \mathbb{R}^{n\times n}$ and $v \in \mathbb{R}^n$. We recognize $Av=\lambda v$ for some scalar $\lambda$ as an eigendecomposition problem. Suppose $\mu \in \mathbb{R}^n$, and let $\odot$ ...
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### Computing eigenvalues of Schrodinger equation with spin

I want to solve a 2-dimensional particle in box problem with two electrons in the quantum well.I would like to take into account spin of electrons and Coulomb interactions to compute singlet and ...
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1 vote
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### Accelerated Inverse Power Method with Rayleigh Quotient

I am considering implementing the accelerated inverse power (AIP) method with Rayleigh quotient to speed up eigendecomposition of a real square symmetric matrix. Halton (1996) gives an example ...
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### Eigenvectors associated to two quasi-degenerate eigenvalues

I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$. Within Matlab enviroment, I use the command ...
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### Eigenvalue with largest imaginary part

Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work. As I understand it, ...
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### Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
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### Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
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### LAPACK sorting eigenvalues differently each time

I'm using LAPACK zgeev routine to get eigenvalues and eigenvectors of a symmetric matrix in C++. Problem is zgeev is being called in a loop but it sorts eigenvalues (and eigenvectors) differently ...
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### Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
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### Correct eigenfunctions of Laplace operator by Finite Differences

I am trying to compute the eigenfunctions of the Laplace operator, i.e. finding $u$ in $$-\nabla^2 u = \lambda u .$$ For now I am trying to do this in 1D, so $$\nabla^2 = \partial_{xx} .$$ I am ...
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