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Class Note for MATH 215 with Professor Dostert at UA 2

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
THE UNIVERSITY Q OF ARIZONA Math 215 Introduction to Linear Algebra Section 41 Introduction to Eigenvalues and Eigenvectors Paul Dostert September 30 2008 Ah Eigenvalue amp Eigenvector Definition Let A be an n gtlt n matrix A scalar A is called an eigenvalue of A is there is a nonzero vector x st Ax Ax Such a vector x is called an eigenvector of A corresponding to A We sometimes refer to the matched pair of an eigenvalue and eigenvector as an eigenpair Note If A is an eigenvalue then for an eigenvector x A AI x 0 This says that eigenvectors x are in the null space of A AI Ex Show that x 12 is an eigenvector of A i 22 and find the corresponding eigenvalue Ex Show that 4 is an eigenvalue of A 21 32 and determine all eigenvectors corresponding to this eigenvalue Ah Eigenspace Let A be an n gtlt n matrix and let A be an eigenvector of A The collection of all eigenvectors corresponding to A together with the zero vector is called the eigenspace of A and is denoted by EA 1 2 1 Ex Show that A 0 is an eigenvalue of A 2 0 2 and find a 1 2 3 basis for its eigenspace What are the other eigenvalues of A In R2 we can think of Ax Ax as saying that Ax and x are parallel So x is an eigenvector iff A transforms x into a parallel vector Ex Find the eigenvalues and eigenvectors of the following matrices AiagivBi 31 geometrically A Finding Eigenpairs Ex Find all eigenvalues and eigenvectors of each of the following matrices 2 2 2 A g 01 7B 0 1 1 4 8 3 Ex Show that the eigenvalues of the lower triangular matrix Alzzi are A a and A d Find the corresponding eigenspaces Ex Show that if A is an eigenvalue of A then Ak is an eigenvalue of A7 If A gt 1 then what happens to the eigenvalues of Ak as k gt oo A Matlab Examples Finding eigenpairs is quite easy in Matlab You simply need to create a matrix then call the eig function It returns two matrices The 15t contains the eigenvectors as columns and the 2quotd is a diagonal matrix that contains the corresponding eigenvalues To find the eigenvalues and vectors of Ali 01 we do A 3 0 8 1 V D eigA d diagD This makes d a vector of the eigenvalues We can verify these are eigenvalues and eigenvectors by computing Ax and Ax AV1 d1V1 AV2 d2V2

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