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Unformatted text preview: Math 334 Lecture #34 Â§ 7.8: Repeated Eigenvalues Finding Linearly Independent Solutions. The eigenvalues of the coefficient matrix in the system, ~x = A~x where A = 1 1 1 3 , are real and repeated: det( A rI ) = 0 â‡’ r 2 4 r + 4 = 0 â‡’ r = 2 , 2 . All of the eigenvectors of A corresponding to r = 2 satisfy 1 1 1 1 Î¾ 1 Î¾ 2 = â‡’ ~ Î¾ = Î± 1 1 where Î± is a nonzero real scalar. [The dimension of the solution subspace of the homogeneous system ( A 2 i ) ~ Î¾ = ~ 0 is 1.] One solution of the system is given by choosing Î± = 1: ~x (1) = 1 1 e 2 t . Is there another solution of the form ~x (2) = ~ Î¾e 2 t that is linearly independent of ~x (1) ? No, because there is only one linearly independent eigenvector of A corresponding to the repeated eigenvalue. [If there were two linearly independent eigenvectors of A corresponding to r = 2, i.e. if the dimension of the solution space of ( A 2 I ) ~ Î¾ = ~ 0 were 2, then there would be two linearly independent solutions of the form ~ Î¾e 2 t for the system.]for the system....
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 Spring '11
 Smith
 Linear Algebra, Eigenvectors, Vectors, Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, Orthogonal matrix, Î±, Î¨

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