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Unformatted text preview: Notes 21: Eigenvalues, Eigenvectors Lecture December 3, 2010 Definition 1. Let F : V → V be a linear map. An eigenvalue for F is a number, λ , real or complex, so that there exists a non-zero vector v ∈ V so that F ( v ) = λv. The vector v is an eigenvector for F with eigenvalue λ. Our goal is to find the eigenvalues, eigenvectors of a given matrix. Let A be an n × n matrix. If λ is an eigenvalue for A , then there is a non-zero vector v ∈ R n so that Av = λv or A ( v- λv ) = 0 or ( A- λI n ) v = 0 . This says that the matrix A- λI has a non-trivial kernel. We know that an n × n matrix has a non-trivial kernel if and only if its determinant is zero. Thus λ is an eigenvalue for the matrix A if and only if det ( A- λI ) = 0 . This gives us an algorithm for finding eigenvalues. Algorithm 1. Let A be an n × n matrix. A number λ ∈ R or C is an eigenvalue for A if and only if det ( A- λI ) = 0. Thus to find the eigenvalues we compute the polynomial det ( A- λI ) and find its roots. The roots are the eigenvalues....
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