# c5s1 - 5.1 Eigenvalues and Eigenvectors 1 Chapter 5....

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5.1 Eigenvalues and Eigenvectors 1 Chapter 5. Eigenvalues and Eigenvectors 5.1 Eigenvalues and Eigenvectors Defnition 5.1. Let A be an n × n matrix. A scalar λ is an eigenvalue of A if there is a nonzero column vector ~v R n such that A~v = λ~v .Th e vector ~v is then an eigenvector of A corresponding to λ . Note. If A~v = λ~v then A~v λ~v = ~ 0andso( A λ I ) ~v = ~ 0. This equation has a nontrivial solution only when det( A λ I )=0 . Defnition. det( A λ I ) is a polynomial of degree n (where A is n × n ) called the characteristic polynomial of A , denoted p ( λ ), and the equation p ( λ ) = 0 is called the characteristic polynomial .

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5.1 Eigenvalues and Eigenvectors 2 Theorem 5.1. Properties of Eigenvalues and Eigenvectors. Let A be an n × n matrix. 1. If λ is an eigenvalue of A with ~v as a corresponding eigenvector, then λ k is an eigenvalue of A k , again with ~v as a corresponding eigenvector, for any positive integer
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## This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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c5s1 - 5.1 Eigenvalues and Eigenvectors 1 Chapter 5....

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