Lecture19 - University of Toronto at Scarborough Department...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 19 Eigenvalues and Eigenvectors Sophie Chrysostomou definition 0.1 Let A be an n × n matrix. A scalar λ is an eigenvalue of A if there is a nonzero vector v ∈ Rn , such that Av = λv. In this case, v is called an eigenvector of A corresponding to the eigenvalue λ. How to Find the Eigenvalues of a Square Matrix definition 0.2 Let A be an n × n matrix. The characteristic polynomial of A is given by p(λ) = |A − λI |. If λ is an eigenvalue of A, then the set Eλ = {x Ax = λx } is called the eigenspace of λ. It contains the zero vector and all the eigenvectors of A corresponding to λ. Note: Eλ = nullspace( A − λI ). 1 20 0 example 0.3 Let A = 1 2 −1 1 3 −2 1. Find the characteristic polynomial of A. 2. Find all of the eigenvalues of A. 3. For each eigenvalue λ of A, find its eingespace Eλ . 2 1 1 −3 0 6 example 0.4 Let A = 2 1 −1 5 1. Find the characteristic polynomial of A. 2. Find all of the eigenvalues of A. 3. For each eigenvalue λ of A, find its eingespace Eλ . 3 ...
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