M334Lec34 - Math 334 Lecture#34 7.8 Repeated Eigenvalues...

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Unformatted text preview: Math 334 Lecture #34 § 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 ⇒ r 2- 4 r + 4 = 0 ⇒ r = 2 , 2 . All of the eigenvectors of A corresponding to r = 2 satisfy- 1- 1 1 1 ξ 1 ξ 2 = ⇒ ~ ξ = α 1- 1 where α is a nonzero real scalar. [The dimension of the solution subspace of the homogeneous system ( A- 2 i ) ~ ξ = ~ 0 is 1.] One solution of the system is given by choosing α = 1: ~x (1) = 1- 1 e 2 t . Is there another solution of the form ~x (2) = ~ ξe 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 ) ~ ξ = ~ 0 were 2, then there would be two linearly independent solutions of the form ~ ξe 2 t for the system.]for the system....
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M334Lec34 - Math 334 Lecture#34 7.8 Repeated Eigenvalues...

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