matlab Diagonalization

# matlab Diagonalization - Eigen Problems and Diagonalization...

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Unformatted text preview: Eigen Problems and Diagonalization Using Matlab An Eigenproblem for a given matrix A requires finding the set of vectors, x , and the scalar numbers λ such that A x = λ x . In other words, we want the vectors which, when operated on by A , are simply multiples of the orginal vector. Matlab allows for easy computation of the eigenvalues and eigenvectors of any square matrix. For example, consider the following Matlab commands: > A = [-3 1 -3; -8 3 -6; 2 -1 2] A =-3 1-3-8 3-6 2-1 2 To find the eigenvalues of A we could use the fact that the eigenvalues, λ satisfy the characteristic equation given by det ( A- λI ) = 0 . Matlab has an easy way of entering this. Simply use the poly command: > p = poly(A) p = 1-1 2 The result says that the characteristic polynomial is: p ( λ ) = λ 3- 2 λ 2- λ + 2 = 0 This can be factored into: ( λ- 1)( λ + 1)( λ- 2) Which gives us the eigenvalues of A directly. If you don’t see the factorization easily, Matlab is equipped to solve the characteristic equation for you using the roots() command, > eigs = roots(p) eigs = 2 1-1 which gives the zeros (eigenvalues) of the polynomial directly. Now we can solve for the eigenvectors of A . For each eigenvalue, we must solve ( A- λI ) x = 0 for the eigenvector x . In Matlab the n × n identity matrix is given by eye(n) . To find the eigenvector associated with λ = 2 we could use: 1 Eigen Problem Solution Using Matlab...
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## This note was uploaded on 01/16/2011 for the course MATH 216 taught by Professor Mckinley,s during the Fall '08 term at Duke.

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matlab Diagonalization - Eigen Problems and Diagonalization...

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