Questions tagged [eigenvalues]

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10
votes
1answer
5k views

Mathematica won't give eigenvectors but Wolfram Alpha will? What am I doing wrong?

If I ask Mathematica to find the eigenvectors and eigenvalues of the matrix: ...
8
votes
1answer
570 views

Finding Eigenvalues for a boundary value problem

I have a 10x10 linearized BVP which I can write as $$\mathbf{y}'(x) = \mathbf{A}(\omega) \mathbf{y}(x)$$ subject to boundary conditions $$\mathbf{B} \cdot \mathbf{y} = \mathbf{0}, \quad x=0 \\ \mathbf{...
6
votes
1answer
458 views

NDSolve eigenvalue problem of bound state

I am trying to solve this eigenvalue problem: \begin{align} \mu \Psi(r) & = -\frac{1}{2}\left ( \Psi^{\prime \prime}(r) + \frac{2}{r} \Psi' (r)\right ) -4\pi \Psi(r) \int _0^\infty dr' r'^2 \frac{...
16
votes
8answers
953 views

Tracking Eigenvalues Through a Crossing

Suppose I have a matrix which depends on some parameter. I want to compute the eigenvalues as a function of this parameter, and then plot them. For example, I may have a matrix representing the ...
6
votes
1answer
459 views

NDEigensystem for structural vibration

Having installed version 11 I thought I would check an old Stack Exchange vibration problem using NDEigensytem. The old problem was Test a wooden board's vibration mode and I think this was before ...
36
votes
2answers
970 views

Eigenvalues broken in Version 12.0

Bug introduced in 12.0. The following code calculates the eigenvalues of a certain complex matrix, which come in pairs of opposite complex numbers. Therefore one can check whether the sum of all ...
4
votes
2answers
135 views

Efficiently find all values of parameter such that any of the eigenvalues of a matrix is equal to 1

I wish to find all values for a parameter such that my matrix has an eigenvalue of 1. Here is an example 16-by-16 matrix with elements depending on the parameter x ...
9
votes
1answer
289 views

Gaining precision/accuracy with NDEigenvalues

See further down for an important note Background I study (one component of) the semi-classical Pauli operator, $$ P_h=-h^2\Delta+ih(-y,x)\cdot\nabla+\frac{x^2+y^2}{4}-h. $$ For this particular ...
3
votes
1answer
132 views

The algebraic solution and numerical solutions for eigenvectors are different. Why?

I find that the algebraic solution for the Eigenvectors of a 3x3 matrix is not correct when compared to the numerical solution. I don't see why ? ...
1
vote
1answer
168 views

Small positive eigenvalues found for a negative definite matrix

I have encountered a problem with Mathematica that I really don't know how to solve. I'm sorry if I am going to be very verbose but I need it to explain the problem properly (and also make the code I'...
0
votes
0answers
127 views

Finding eigenvalues of a differential operator

I am trying to get the eigenvalues of the following differential operator $$L\psi(r) = -f \partial_r (f \partial_r \psi(r)) + V \psi(r)$$ which must satisfy (obviously) $$L \psi(r) = \omega^2 \psi(...
13
votes
6answers
723 views

Solving this challenging ODE

Consider the ODE: $$w^{(4)}(x) + (L-x)w''(x) - w'(x) = 0 $$ with some of the following boundary conditions: free: $w'' = 0$, $w'''=0$, clamped: $w = 0$, $w'=0$, pivot: $w = 0$, $w''=0$. Two such ...
11
votes
2answers
383 views

Making an interactive visualization of the eigenvectors of two-dimensional matrices

I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors. How can this sort of visualization be ...
8
votes
3answers
596 views

How to set interface conditions for optical waveguide in NDEigensystem?

I have been working on waveguide mode analysis using FEM in Mathematica for a week, but I haven't succeeded until now. The optical fiber-like waveguide is featured with different refractive index in ...
7
votes
5answers
511 views

Spectral problem for differential operator

I want to find numerically eigenvalue and eigenfunctions some nontrivial differential operator. But I can not find, how to do it in Mathematica. For the sake of simplicity let us discuss the trivial ...
6
votes
1answer
113 views

NDEigenvalues complains not Hermitian with large dimension differential operator

The following snippet calculates eigenvalues and eigenfunctions of a null operator (just as an example): ...
4
votes
1answer
248 views

Finding the eigenvalues (diagonalizing) of a block-diagonal matrix

I have a large $2^N \times 2^N$ matrix. It is the exact Hamiltonian of a spin chain model which I have generated with code I wrote in Fortran. The code block diagonalizes the Hamiltonian into constant ...
16
votes
0answers
453 views

Wrong eigenvalues from a sparse matrix

Bug introduced after 5.0, in or before 8.0 and persisting through 12.0. I notice in the following example that wrong smallest 2 eigenvalues are resulted if calculating from a sparse matrix. But it ...
6
votes
2answers
169 views

Solving eigenvalue BVP with an interface

I have a boundary-value problem, that is defined over two adjacent regions with an interface in the middle, that contains an eigenvalue $\lambda$. The boundary conditions and the equations are ...
4
votes
2answers
7k views

How to normalize a list of eigenvectors?

Here is a simple eigenvector problem solution m = {{2, Sqrt[15]}, {Sqrt[15], 4}}; v = Eigenvectors[m] However, the list of vectors v is not normalized. The ...
4
votes
1answer
135 views

Numerically computing the eigenvalues of an infinite-dimensional tridiagonal matrix

I have one infinite dimensional tridiagonal matrix whose eigenvalues I have to compute. How can that be done numerically using Mathematica? Let me expose the concrete case I want to do it. I shall ...
2
votes
2answers
105 views
6
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2answers
228 views

Unable to evaluate Eigenvalues and Eigenvectors for a matrix

I have the following 3X3 matrix M and I wish to find its eigenvectors and eigenvalues ...
3
votes
0answers
53 views

Interface points of NDEigensystem

When solving an eigenvalue problem with "NDEigensystem", e.g. a 1D Eigenvalue problem with the interval composed of different materials, which should be solved by the pure numerical method such as FEM,...
3
votes
2answers
314 views

Why ODE's naive finite difference matrix works well for different boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. The naive way to write down the matrix of the differential operator is like the following, ...
3
votes
1answer
181 views

Discontinuities in eigenvalues plotting with Plot3D

I have a 6x6 matrix, depending on 2 variables kx and ky and defined as the sum of the two following matrices (...
1
vote
1answer
293 views

How to collect eigenvectors corresponding to only positive eigenvalues?

Let us consider a matrix of order $n \times n$ with $n/2$ positive and $n/2$ negative eigenvalues. How to collect $n/2$ eigenvectors corresponding to positive eigenvalues in a matrix of order $n \...
1
vote
2answers
76 views

Eigenvalues error: “The method ”Banded“ accepts only sparse matrices with elements that are machine-real or machine-complex numbers”

I'm having one issue with the Eigenvalues function in some code priorly discussed here. There the example is tridiagonal, but here, let us consider this simple ...
0
votes
1answer
135 views

Using multiple boundary conditions with NDEigensystem

I'm quite new to Mathematica and to Stack Exchange so I apologise if this question has already been answered. I've recently been trying to solve a partial differential equation to find the ...