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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 =3 138 36 21 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 = 11 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 11 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.
 Fall '08
 Mckinley,S
 matlab, Vectors, Scalar

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