9 - Eigenvalues and Eigenvector

9 - Eigenvalues and Eigenvector - 1 Math 1b Practical...

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1 Math 1b Practical — Eigenvalues and eigenvectors, I February 9, 2011 Let A be a square matrix. A (right) eigenvector with corresponding eigenvalue λ is a nonzero (column) vector e so that A e = λ e . (1) We say λ is an eigenvalue of A when (1) holds for at least one nonzero vector e . You will be asked to ±nd the eigenvalues and eigenvectors for various matrices. This means ±nd all eigenvalues, and for each eigenvalue λ , describe all corresponding eigenvec- tors. Note that λ is an eigenvalue of A if and only if A λI is singular, since (1) is equivalent to ( A λI ) e = 0 . This in turn is equivalent to det( λI A ) = 0. The polynomial det( λI A )=( 1) n det( A λI ) is caled the characteristic polynomial of A ; the zeros of the charateristic polynomial are the eigenvalues of A . G ivenane igenva lue λ , the subspace U λ = { e : A e = λ e } = null space of A λI is called the eigenspace of A corresponding to λ , and its dimension is called the geometric multiplicity of λ as an eigenvalue of A . The nonzero elements of U λ are exactly the eigen- vectors with corresponding eigenvalue λ ,and 0 . To describe the eigenvectors corresponding to λ , we may give a basis for the eigenspace U λ .
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This note was uploaded on 11/10/2011 for the course MA 1B taught by Professor Aschbacher during the Winter '08 term at Caltech.

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9 - Eigenvalues and Eigenvector - 1 Math 1b Practical...

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