Solving Eigenvalue / Eigenvector Problem

In summary: Your name]In summary, the problem discussed is from a text on FEA and involves solving an eigenvalue/eigenvector problem. The goal is to show that L_n = (2n-1)pi/(2a) and the solution is u(r,T) = sum[a_n r^(L_n) (cos(L_n*T)+(-1)^n sin(L_n*T)] for n = 1 to infinity. The user has attempted to solve the problem using MATLAB but is stuck. They are trying to transform the problem into the form (A-LB)x = 0 and have used the "eig" function to find the eigenvalues and eigenvectors of matrix A. They hope this will help them solve
  • #1
paxprobellum
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Homework Statement


The problem is from a text on FEA, but I've "solved" the problem down to an eignenvalue/eigenvector problem. The point is to show that L_n = (2n-1)pi / (2a) and that the solution u(r,T) = sum [ a_n r^(L_n) ( cos (L_n * T) + (-1)^n sin (L_n * T) ] for n = 1 to infinity.

L = lambda, T = theta

Homework Equations



r^L (c1 cos(aL/2) + c2 sin(aL/2) ) = 0
L r^L (c1 sin(aL/2) + c2 cos(aL/2) ) = 0

The Attempt at a Solution



I want to get it into the form (A-LB)x = 0 to solve it using MATLAB (sptarn, e.g.) but I don't know how. So I moved on to say that if Ax = 0 and there are nontrivial solutions, than the determinant of A is zero as:

det( [cos(aL/2) sin(aL/2) ; sin(aL/2) cos(aL/2)] ) = 0

but that gives me aL/2 = 0 (cos^2 + sin^2 = 1), which is not the answer. So I'm a bit stuck.
 
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  • #2


Thank you for sharing your progress on the problem. It seems like you have made some good progress in reducing the problem to an eigenvalue/eigenvector problem. To solve it using MATLAB, I would suggest using the "eig" function, which will give you the eigenvalues and eigenvectors of a given matrix.

In this case, your matrix A would be:

A = [cos(aL/2) sin(aL/2); sin(aL/2) cos(aL/2)]

And your vector x would be:

x = [c1; c2]

To solve for the eigenvalues and eigenvectors, you can use the command:

[eig_vec, eig_val] = eig(A)

This will give you the eigenvectors (eig_vec) and eigenvalues (eig_val) of matrix A. From there, you can use the eigenvalues to solve for L_n and the eigenvectors to solve for the coefficients a_n.

I hope this helps. Good luck with your problem!
 

1. What is an eigenvalue and eigenvector?

An eigenvalue is a scalar value that represents the amount by which a particular eigenvector is stretched or compressed when a linear transformation is applied. An eigenvector is a vector that does not change its direction when a linear transformation is applied.

2. Why is solving the eigenvalue/eigenvector problem important?

Solving the eigenvalue/eigenvector problem is important because it allows us to understand the behavior of a linear transformation. It is also used in many fields, such as physics, engineering, and computer graphics.

3. What is the process for solving the eigenvalue/eigenvector problem?

The process for solving the eigenvalue/eigenvector problem involves finding the eigenvalues and corresponding eigenvectors of a given matrix. This can be done by first finding the characteristic polynomial of the matrix, then solving for the eigenvalues, and finally finding the corresponding eigenvectors.

4. Can the eigenvalue/eigenvector problem be solved for any matrix?

No, the eigenvalue/eigenvector problem can only be solved for square matrices. This means that the number of rows is equal to the number of columns in the matrix.

5. What are some real-world applications of the eigenvalue/eigenvector problem?

The eigenvalue/eigenvector problem is used in many real-world applications, such as image and signal processing, population dynamics, quantum mechanics, and machine learning. It is also used in studying the stability of systems, such as in economics and engineering.

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