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Lecture

# Eigenvalues, eigenvectors, and diagonalization

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University of Toronto Scarborough

Mathematics

MATA23H3

Sophie Chrysostomou

Winter

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Eigenvalues and Eigenvectors Denition: Let A be an n n matrix. A scalar is an eigenvalue of A if there is a n nonzero vector v R , such that Av = v. In this case, v is called an eigenvector of A corresponding to the eigenvalue . How to Find the Eigenvalues of a Square Matrix Denition: Let A be an n n matrix. The characteristic polynomial of A is given by p() = A I. If is an eigenvalue of A, then the set E {x Ax = x} is called the eigenspace of . It contains the zero vector and all the eigenvectors of A corre- sponding to . Note: E =nullspace(A I ). 1 2011 by Sophie Chrysostomou www.notesolution.com 2 0 0 Example: 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 of A, nd its eingepace E . 2 2011 by Sophie Chrysostomou www.notesolution.com

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