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Unformatted text preview: MAT 1302  Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2010 Lecture 20 Alistair Savage (uOttawa) MAT 1302  Mathematical Methods II Winter 2010 Lecture 20 1 / 29 Announcements Fourth Assignment: Posted on Virtual Campus. Due at 8:30am, April 6. Weve already covered the material necessary for the assignment. Teaching evaluations: Beginning of class on Tuesday, March 30. Please come to class and be on time. Review class (April 9): The last class is open for review. We can go over theory, do extra examples, or go through an old exam. Think of topics that youd like to go over well collect requests next week. Alistair Savage (uOttawa) MAT 1302  Mathematical Methods II Winter 2010 Lecture 20 2 / 29 Eigenvectors, eigenvalues, and eigenspaces Definition (eigenvectors and eigenvalues) Suppose A is a square matrix. If ~ x is a nonzero vector and is a scalar such that A ~ x = ~ x , ~ x 6 = ~ then is an eigenvalue of A , and ~ x is an eigenvector of A (an eigenvector corresponding to the eigenvalue ). If is an eigenvalue, then the set of solutions to A ~ x = ~ x (or ( A I ) ~ x = ~ 0) is the eigenspace corresponding to . Note: Since the eigenspaces are null spaces of A I , they are subspaces. Alistair Savage (uOttawa) MAT 1302  Mathematical Methods II Winter 2010 Lecture 20 3 / 29 Recap: finding eigenvalues and eigenvectors/eigenspaces Procedure for finding eigenvalues, eigenvectors and eigenspaces 1 To find the eigenvalues, find the solutions to the characteristic equation det( A I ) = 0 . 2 For each each eigenvalue, solve the equation ( A I ) ~ x = ~ to find the corresponding eigenspace. 3 The nonzero vectors in each eigenspace are the eigenvectors corresponding to the given eigenvalue. Alistair Savage (uOttawa) MAT 1302  Mathematical Methods II Winter 2010 Lecture 20 4 / 29 Diagonal matrices Recall: a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal. Examples: 1 0 0 0 6 0 0 0 1 2 2 0 0 3 0 0 4 0 0 5 Diagonal matrices are easy to work with for many reasons: They are both upper and lower triangular. Their eigenvalues are just the entries on the diagonal. Their determinant is simply the product of the entries on the diagonal. Easy to see if they are invertible (and find inverses). Easy to multiply (just multiply corresponding diagonal entries). Alistair Savage (uOttawa) MAT 1302  Mathematical Methods II Winter 2010 Lecture 20 5 / 29 Example Powers of arbitrary matrices can be difficult to compute: A = 2 10 12 1 3 1 6 3 4 8 5 4 7 5 7 4 , A 4 is a lot of work to compute....
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 Spring '11
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