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Unformatted text preview: The Eigenvector/Eigenvalue Method We first recall the following from linear algebra: Definition Let A be an n × n matrix. A vector v such that A v = λ v for some scalar λ is called an eigenvector (EV) of A with eigenvalue (ev) λ . Before we discuss how these objects can be applied to solving the homogeneous equation ˙ y = A y , we review the process for finding the eigenvectors and eigenvalues of a given matrix A . Note that Av = λ v m ( A λI ) v = 0 ⇓ det( A λI ) = 0 where I is the n × n identity matrix (the matrix such that every entry on the main diagonal is 1 and every other entry is zero). Since det( A λI ) is a polynomial in the variable λ , we need only find the zeros of this polynomial to find all the eigenvalues of A . To find a corresponding eigenvector, we must solve the system ( A λI ) v = 0 for the components of v . I Example Find the eigenvalues and corresponding eigenvectors of the matrix A = 4 2 1 1 . Solution The eigenvalues are the roots of the polynomial det( A λI ) . We compute A λI = 4 2 1 1 λ 1 0 0 1 = 4 λ 2 1 1 λ Then we compute det( A λI ) = (4 λ )(1 λ ) ( 1)(2) = λ 2 5 λ + 6 Setting this to zero we have 0 = λ 2 5 λ + 6 = ( λ 2)( λ 3) so the eigenvalues are λ 1 = 2 and λ 2 = 3 . To find the eigenvectors v = v 1 v 2 corresponding to λ 1 = 2 , we need to solve ( A λ 1 I ) v = 0 that is, 4 2 1 1 2 1 0 0 1 v 1 v 2 = . ⇓ 2 2 1 1 v 1 v 2 = ⇓ 2 v 1 + 2 v 2 = 0 v 1 v 2 = 0 Since the second equation is just a scalar multiple of the first, this system has infinitely many solutions of the form v 1 = v 2 ....
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.
 '08
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 Algebra, Scalar

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