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eigen vec& val - Harvey Mudd College Math Tutorial...

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Harvey Mudd College Math Tutorial: Eigenvalues and Eigenvectors We review here the basics of computing eigenvalues and eigenvectors. Eigenvalues and eigenvectors play a prominent role in the study of ordinary differential equations and in many applications in the physical sciences. Expect to see them come up in a variety of contexts! Definitions Let A be an n × n matrix. The number λ is an eigenvalue of A if there exists a non-zero vector v such that A v = λ v . In this case, vector v is called an eigen- vector of A corresponding to λ . Computing Eigenvalues and Eigenvectors We can rewrite the condition A v = λ v as ( A - λI ) v = 0 . where I is the n × n identity matrix. Now, in order for a non-zero vector v to satisfy this equation, A - λI must not be invertible. That is, the determinant of A - λI must equal 0. We call p ( λ ) = det( A - λI ) the characteristic polynomial of A . The eigenvalues of A are sim- ply the roots of the characteristic polynomial of A . Otherwise, if A - λI has an inverse, ( A - λI ) - 1 ( A - λI ) v = ( A - λI ) - 1 0 v = 0 . But we are looking for a non-zero vector v .
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