Questions tagged [eigenvalues]

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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 ...
1k views

Complex eigenvalues from a sparse Hermitian matrix

Bug introduced in 9.0 or earlier and persisting through 12.0. I notice in the following example that wrong complex eigenvalues are resulted if calculating from a Hermitian sparse matrix, which should ...
1k views

Eigenvalues broken in 11.2?

Bug introduced in 11.1.0 and fixed in 11.3.0 The code ...
1k views

How to check if a vector is an eigenvector of a matrix using mathematica?

Here is a vector $$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$ Here is a matrix $$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$ Is there a simple way to determine ...
870 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 ...
442 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 ...
833 views

An ODE system easily polluted with spurious eigenvalues

I tried solving the eigenvalue problem of a 1st-order ODE system (see the code below) with NDEigenvalue. (One option I found in it seems to be ...
247 views

Specifying initial vector for finding Eigenvectors using Arnoldi method

I am trying to speedup the calculation of eigenvalues, given that I have good guesses for the eigenvectors. From what I know of Arnoldi/Lanczos, my good guesses should be helpful. Unfortunately, I am ...
704 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 ...
372 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 ...
657 views

What algorithm is Mathematica using to find the smallest eigenvalue so quickly?

My question is what kind of black magic is Mathematica doing to obtain the correct answer so quickly compared to other programming languages? Details: I've written a Mathematica notebook to find the ...
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: ...
568 views

Nonlinear ODE eigenvalue problem

How does one find eigenvalues $\lambda$ of the following problem? $$\frac{\mathrm{d}^2 u}{\mathrm{d}x^2} = \lambda \left( -u + u^2 \right),$$ $$u(0) = u(1) = 0.$$ Can this be tackled by ...
372 views

NDEigenvalues vs. FindRoot for finding the eigenvalues of Airy equation?

Say I am trying to find the first 5 eigenvalues of the differential equation $f''(x)=\lambda x f(x)$, on the interval [-1,0], with boundary conditions $f(-1)=f(0)=0$. I will try to do this 3 ways, ...
330 views

How to solve this 2nd-order ODE with singularity?

I tried solving the eigenvalue problem of a 2nd-order ODE $$[b^2(k-2)^2y^2-2b(k-2)(1+2ky)+4k^2+b^2(k-2)3y]f(y) \\- 3b(3by-2)f'(y)\\-(3by-2)^2f''(y)=\lambda f(y)$$ with ...
275 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 ...
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Using NDEigensystem to solve the Mathieu equation

To be able to apply the differential equation capabilities of Mathematica to my graduate thesis, I am trying to apply NDEigensystem to an eigenproblem whose solution I know, but I am having some ...
107 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): ...
191 views

Eigensystem returns vectors which are not eigenvectors

Short synopsis: for a specific family of sparse matrices, the eigensolver seems to be unstable (kernel quitting) for certain examples, and when it works it seems to consistently return vectors which ...
163 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 ...
438 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{...
450 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 ...
126 views

Inconsistency in eigenvalues of matrices in a specific form (sparse & non-Hermitian)

Suppose one has a non-Hermitian sparse matrix defined as below ...
202 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 ...
I have the following matrix (the DFT Matrix for N = 3) $$W = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & e^{-\frac{i 2 \pi}{3} } & e^{\frac{i 2 \pi}{3} } \\ 1 & e^{\frac{... 2answers 895 views Noise in Eigenvalues plot I am trying to Plot Eigenvalues of a Hamiltonian, but I am getting noisy plot, which is incorrect. Here is the code. ... 2answers 147 views Lowest Magnitude Eigenvalues of Large Sparse Matrices I am trying to find the first three lowest eigenvalues of large sparse matrices of size range 10^3 - 10^5. The matrices depend on some parameter x, so I first construct the matrices and then use ... 2answers 113 views How to show just one function from a stored plot? Q: Is there a general way to remove particular functions from a previously stored call to a plot function? Here is a specific example: ... 1answer 146 views Finding the orthogonal diagonalizing similarity of a symmetric matrix I'm aware that there are some questions similar to this here, but none that could solve my problem. So, I have to diagonalize a symmetric symbolic matrix m (to be seen below) and obtain the ... 1answer 575 views How to plot the eigenvalues of a parametric matrix efficiently? I was wondering how can I set the variable type of matrix elements to be real. The problem is, I creat a variable-dependent matrix as follows and I get the eigenvalues now I want to plot the ... 1answer 361 views How to plot complex eigenvalues of a matrix? I have a matrix , for instance, like this : matrix[a_ ] := {{0, a}, {-a, 1}}; Eigenvalues[matrix[a]] and this give the eigenvalues that depends on ... 1answer 239 views How to specify both DirichletCondition and NeumannValue on DEigenvalues? Ok, I give up :) I am trying to verify the eigenvalues for heat PDE in 1D, when left end has homogeneous DirichletCondition but the right end is insulated, so it ... 1answer 135 views Speed up selecting positive eigenvalues repeatedly I have a smallish (e.g. 2x2 or 4x4, but ideally up to 10x10) non-symmetric square matrix \mathbf{A}(x). I need to define a function f(x) which is the sum of the eigenvalues with positive real part ... 1answer 256 views Eigenvalues of large symmetric matrices When I try to compute the eigenvalues of the adjacency matrix of a very large graph I get, what can be charitably described as, garbage. In particular, since the graph is four-regular, the eigenvalues ... 1answer 140 views How to control DifferenceOrder in NDEigenvalue for an ODE? I am trying to solve the eigenvalue problem of a 1st-order ODE system using NDEigenvalue. It should be finite difference method for ODE. And I want to tune the the ... 1answer 434 views How to use DEigensystem with periodic boundary conditions on the derivative? Since I found I can use DEigensystem to obtain eigenvalues and eigefunctions for differential operators, I am using it to verify solution to some of my HW's. In ... 2answers 189 views Checking if a symbolic symmetrical Matrix is negative definite I have the following problem finding the value ranges for the parameters of a symbolic symmetrical matrix in order to make it negative definite: The matrix I'm talking about looks as follows ... 1answer 109 views Coloring specific set of points in a plot I have a large set of points.I get two bands of points in the graph which merge upon choosing suitable value of the parameters E1 and E2. My question is how to color these two sets of points ... 1answer 69 views How to change the default normalization for NDEigensystem? I'm currently using NDEigensystem to solve a PDE that describes a particle travelling on a hyperbolic (negatively curved) surface. However, the eigenfunctions that are returned by NDEigensystem are ... 0answers 77 views Bug for Cubics -> True in Eigenvalues? Let's consider the simple code ... 2answers 915 views Two matrices that are not similar have (almost) same eigenvalues [closed] I have two matrices$$ A=\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \text{ and } \quad B=\begin{pmatrix} d & e & f \\ d & e &...
I have a linear equation where the unknown is a matrix: $$\sum_{pq} T_{ijpq}X_{pq} = \lambda X_{ij}$$ This is an eigenvalue problem for the tensor $T_{ijpq}$. Mathematica ...