Questions tagged [eigenvalues]

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4
votes
1answer
155 views

Solve an ODE with parameters in a boundary condition

Consider the ODE: ode = y''''[x] - 2*k^2*y''[x] + k^4*y[x] == I*k*a*((2*x - x^2 - c)*(y''[x] - k^2*y[x]) + 2*y[x]); in which a...
0
votes
2answers
75 views

Trying to to replace Root expressions from the output of Eigenvalues by the explicit forms

When I calculate the eigenvalues of the following matrix (H) by using Eigenvalues, I get complex expressions with ...
1
vote
2answers
38 views

Eigenvalues function not computing correct imaginary part

I have the following matrix, with a free variable kx. ...
3
votes
1answer
79 views

Diagonalizing a symbolic matrix

I am trying to diagonalize the following matrix \begin{equation} \left(\begin{array}{cccc} { \frac{1-K\left(x_{2}^{2}+x_{3}^{2}\right)}{1-K|x|^{2}}} & { \frac{K x_{1} x_{2}}{1-K |x|^{2} }} & { ...
40
votes
3answers
1k 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 ...
0
votes
0answers
38 views

Finding the Eigenvalues for big matrices

To start I have the following matrix (not so big) I want to find the Eigenvalues in terms of kx and ky. I tried to use the classical ...
0
votes
1answer
106 views

Plotting eigenvalues

I am trying to reproduce figure 10 in the linked paper. The authors have used Matlab bvp4c function to find multiple solutions for the system of ODE and then carried out the stability. Here is my ...
1
vote
2answers
63 views

Problems with spectral decompositon of a $4 \times 4$ matrix [duplicate]

Given a matrix $M$ with eigenvalues $\lambda_1, \lambda_2, \lambda_3,\lambda_4$ and the corresponding eigenvectors $|v1\rangle,|v2\rangle,|v3\rangle,|v4\rangle$. One can write $ M = \lambda_1 |v1\...
5
votes
2answers
109 views

Eigenvalues of a fourth-order ODE

Consider the following ODE for $y(x)$ over $x\in\left[0,\frac{1}{2}\right]$ with an eigenvalue $\lambda$ $\qquad 2x\,y''''+ 4y'''=\lambda\, y''$ The boundary conditions at $x=\frac{1}{2}$ are $y'\...
3
votes
1answer
72 views

Proving the positive semidefiniteness of a 6X6 symbolic matrix

Specifically, I want to check the positive semidefiniteness of the following 6X6 symbolic matrix ...
2
votes
1answer
91 views

Eigenvalue decomposition of a density matrix not reproducing original density matrix

A density matrix $\rho$ in quantum mechanics is defined as any self adjoint and positive semidefinite matrix with a trace or 1. It can be expanded into sets of pure states such that $\rho=\sum_{i}p_{...
1
vote
1answer
69 views

Problem with QRDecomposition [closed]

I was using Mathematica for the QR decomposition method. But I got strange results. I wanted to find eigenvalues of a matrix, say ...
2
votes
2answers
98 views

Different result from utilizing of eigenvalues and eigenvectors commands

I have a matrix as a function of a parameter like ...
0
votes
0answers
20 views

Using Eigensystem to solve eigenvalue problem [duplicate]

I'm trying to solve an eigenvalue problem of the form: $\textbf{A}\vec{x}=\lambda \textbf{B}\vec{x}$, where A and B are square matrices, $\lambda$ are the eigenvalues and $\vec{x}$ the eigenfunctions. ...
4
votes
2answers
224 views

Help with coding a matrix

I have a $n \times n$ matrix $A$ with a full set of eigenvalues $\lambda$ including repetitions. I want to create the following $i \times i$ matrix: $$\left(\sum_{a=2}^i (a-1) |a-1⟩⟨a| \right) + \...
3
votes
1answer
81 views

Finding eigenvalues of the Laplacian on solenoidal (divergence-free) vector fields

In Mathematica it is easy to find eigenvalues of the Laplacian in simple cases. For example, on $\Omega\in \mathbb{R}^2$: ...
5
votes
1answer
92 views

Eigenvalues and numerical eigenfunctions for similar differential operators

I am looking to numerically approximate the eigenvalues and eigenfunctions for a differential operator I am working with, assuming $\pi$ periodic boundary conditions. Namely, I define the function $...
1
vote
2answers
155 views

Finding specific eigenvalues

Given an $n\times n$ matrix $Q$ (with e.g. $n\approx10^4$) I am only interested in the 3rd smallest eigenvalue of $Q,$ and not the entire spectrum (assume all eigenvalues are real, e.g. a Hermitian ...
2
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0answers
29 views

What kind of performance should I expect out of Eigensystem using FEAST?

I'm numerically solving a time-independent Schrödinger equation using Eigensystem's FEAST method. It takes a lot longer than I ...
1
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0answers
113 views

Somewhat Singular Sturm-Liouville Equation (With Edition)

I am trying to solve the following Sturm-Liouville equation (i.e., plot the eigenfunctions and calculate the eigenvalues): $$\frac{d}{dx}\left(x²\frac{d}{dx}\right)f(x) + 2f(x) = -\lambda x²f(x)\,,$$ ...
4
votes
1answer
71 views

“Arnoldi” method for Eigenvalues inside FindRoot

I'm trying to implement a function which, given a matrix with one free parameter, would return the value of the parameter at which the lowest eigenvalue of the matrix is equal to a certain number. ...
0
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0answers
34 views

Trying to extract the Eigen values and Eigen vector of a matrix

I have a Matrix A, which is a function of ω. I wanted to find the eigenvalues and eigenvectors of this matrix, how to do it. I have used the EigenSystem function ...
2
votes
2answers
122 views

Finding eigenvectors of a differential operator

How can I find the eigenvalues and eigenvectors(numerically) of the below matrix equation: $ \qquad \hat{A}\left({\begin{array}{c} y_1(x,\theta)\\ y_2(x,\theta) \\ \end{array} } \right)= ...
3
votes
2answers
70 views

Checking NDEigensystem Results

I'm looking to verify the output of a call to NDEigensystem. I'm doing this by plotting the operator acting on the Interpolating Function outputs versus the ...
2
votes
0answers
61 views

Block diagonalizing a complex anti-symmetric matrix

I am going to evaluate the block diagonal form of few skew-matrices. When matrix elements are real I can simply follow the approach suggested in this thread which I have implemented that as ...
4
votes
3answers
198 views

Eigenvectors in the limit $ \mu\rightarrow 0 $ are not the same as eigenvectors when setting $ \mu=0 $ from the beginning

I would like to find the eigenvectors of a matrix and see what the eigenvectors look like in the limit of $ \mu\rightarrow 0 $: ...
2
votes
0answers
105 views

Why doesn't NDEigenSystem give smooth eigenfunctions? [closed]

I'm looking for smooth solutions of the 1D Helmholtz equation $\left[\frac{d^{2}}{dx^{2}}+k_{0}^{2}\epsilon(x)\right]\phi=0$ with homogeneous Dirichlet boundary conditions, where the permittivity $\...
0
votes
1answer
56 views

Remove +0Is from Root function

I'm trying to find the zeros of the eigenvalues (functions of $k_x$) of a self-adjoint $4\times4$ matrix H: ...
3
votes
1answer
49 views

Trace change of one single value in a list

I have a list of eigenvalues, say: list1 = {10., 9., 9., 8.5, 7.5, 6.5, 6.1, 5.6, 4.5, 4., 4., 3.8, 3., 3., 1., 1., 1., 0.8, 0.5, 0.5} After slightly modifying ...
3
votes
1answer
106 views

Problem with complex eigenvalues in periodic Sturm-Liouville problem

I'm having trouble using NDEigenvalues to obtain the first few eigenvalues for a differential operator on the circle of radius one-half. $\qquad Lf(x) = f''(x)+ (-...
1
vote
0answers
34 views

Getting a non zero determinant of matrix R, when the Rank of R is not equal to Dimension of R

I have a square matrix whose dimensions is 9 cross 9, when I extract the rank of the matrix R, I am getting rank as 6. I have constructed R matrix by minimizing the Lagrangian Lg with respect to a[1].....
0
votes
0answers
27 views

Sequential production of Eigenvectors?

I have to deal with very large matrices in Mathematica (dimensions $10^4\times10^4$ at least). Obtaining the eigenvalues of these matrices is not so difficult since, it is not memory intensive or ...
0
votes
1answer
90 views

How can I get the Eigen system of a certain matrix? [closed]

How can I get Eigen system of c, where c = a - iota * b? Please help me to find the Eigen system in a nice form. ...
4
votes
1answer
72 views

find a maximum parameter for a range of target eigenvalues as a function of matrix dimension

I have a symbolic tridiagonal matrix of this form ...
4
votes
2answers
946 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 &...
8
votes
0answers
277 views

DEigenvalues and NDEigenvalues return different values

In the following example, DEigenvalues and NDEigenvalues return different results despite having identical arguments. Does anyone know why? (I use Mathematica 11.3) ...
3
votes
2answers
772 views

Problem with Eigenvectors

When I want to calculate eigenvectors of the following matrix in Mathematica the only answer it gives me is zero vector, anybody knows how to fix this? here's my matrix : \begin{equation} X=\left(\...
3
votes
0answers
65 views

Derivative of eigenvalues

I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ...
5
votes
2answers
118 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: ...
7
votes
1answer
133 views

How to solve this 1st-order linear ODE system with a few discrete eigenvalues?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities $$ -i\partial_xu(x)+f^*(x)v(x)=\lambda u(...
3
votes
1answer
321 views

1st-order linear ODE system gives inaccurate/biased solutions

Consider an ODE eigensystem $$ t(y+\frac{1}{s})a(y)+[(q+\frac{1}{2}+\frac{s}{2}y)+s(y\partial_y+\frac{1}{2})]b(y)=\lambda a(y)\\ t(y+\frac{1}{s})b(y)+[(q+\frac{1}{2}+\frac{s}{2}y)-s(y\partial_y+\frac{...
0
votes
0answers
82 views

How to draw mode vectors of two degrees of freedom

Here is two degrees of freedom system. *And the mode vectors of this system is ({{1, 1},{1, -1}}) *And the mode shape will be expressed like this... And the ...
0
votes
0answers
134 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(...
0
votes
0answers
41 views

Some issues with DEingesystem

I would like to solve (get its eigenvalues/vectors) the Sturm-Liouville problem, for the following differential operator: $L =\partial_{r} \partial_{r} \psi(r)$. Also, I would like to impose the ...
33
votes
1answer
2k 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 ...
0
votes
0answers
53 views

Eigenvectors of Hermitian matrices [duplicate]

I asked a similar question in the physics stack exchange, but realized my question is probably more suited here. For any Hermitian matrix $H = H^{\dagger}$ we can write $H = P DP^{\dagger}$ where $P$ ...
1
vote
1answer
148 views

Eigenvalue problem

my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation $\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$ I have a problem in calculating the ...
0
votes
0answers
34 views

Ordering of Eigenvectors [duplicate]

I am interested in computing the derivatives of the eigenvalues of a certain $n\times n$ Hermitian matrix $M(t)$. I know I can do this easily since I know the exact expression for $\dot{M}$, and the ...
9
votes
3answers
624 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 ...
5
votes
2answers
218 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 ...

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