eigen - Handout on the eigenvectors of distinct eigenvalues...

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Handout on the eigenvectors of distinct eigenvalues 9/30/04 This handout shows, f rst, that eigenvectors associated with distinct eigenvalues of an abitrary square matrix are linearly indpenent, and sec- ond, that all eigenvectors of a symmetric matrix are mutually orthogonal. First we show that all eigenvectors associated with distinct eigenval- ues of an abitrary square matrix are mutually linearly independent: Suppose k ( k n ) eigenvalues { λ 1 , ..., λ k } of A are distinct, and take any corresponding eigenvectors { v 1 , ..., v k } , de f ned by v j 6 =0 ,Av j = λ j v j for j =1 , ..., k. Then, { v 1 , ..., v k } are linearly independent. First consider two such eigenvectors. Suppose we have eigenvalue λ with eigenvector v , and eigenvalue µ with eigenvector w , λ 6 = µ .W e will show that αv + βw =0 α = β =0 , implying that v and w are linearly independent. So suppose we have
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eigen - Handout on the eigenvectors of distinct eigenvalues...

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