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Unformatted text preview: Finding Eigenvectors Some Examples General Information Eigenvalues are used to find eigenvectors. The sum of the eigenvalues is called the trace. The product of the eigenvalues is the determinant of the matrix. An EIGENVECTOR of an n x n matrix A is a vector such that , where v is the eigenvector. Av v λ = Eigenvectors An eigenvector is a direction for a matrix. What is important about an eigenvector is its direction. Every square matrix has at least one eigenvector. An n x n matrix should have n linearly independent eigenvectors. Homogeneous Linear Systems Distinct Eigenvalues 1 2 3 1 1 1 gives, after solving det ( ) 0, 1, 0, 2 1 1 X X A I λ λ λ λ ′ =  = = = = 1 1 1 1 Consider 0. 1 1 1 1 A I λ λ =  = → → N ote 0 (secon d row ) and 0 .1 If 1, on e eigenvector is 0 . 11 T he generalized eigenvecto r m ay be w ritten 0 , . 1 y x z x z z s s = + = → =  = ∈ ℜ Distinct Eigenvalues (cont) For so that x=0 (row 3) , z=0 (row 1), y can be anything. If y =1, one eigenvector is The generalized eigenvector may be written 0 0 1 0 1, 0 0 0 0 0 1 0 0 0 A I λ λ =  = → 1 1 , r r ∈ ℜ Distinct Eigenvalues (cont) For so that y =0 (row 3) , x+z=0 ....
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 Spring '11
 Prof.Ramachandran
 Linear Algebra, Eigenvalues, Eigenvalue, eigenvector and eigenspace, eigenvector, Generalized eigenvector

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