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M334Lec32

# M334Lec32 - Math 334 Lecture#32 Â 7.6 Complex Eigenvalues...

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Unformatted text preview: Math 334 Lecture #32 Â§ 7.6: Complex Eigenvalues Finding Real Solutions. Suppose that a constant n Ã— n matrix A has a complex- conjugate pair of eigenvalues: r 1 = Î» + iÎ¼, r 2 = Î»- iÎ¼. If ~ Î¾ (1) is a complex eigenvector corresponding to r 1 , then a complex eigenvector corre- sponding to r 2 = Â¯ r 1 is ~ Î¾ (2) = Î¾ (1) (the component-wise complex conjugate): ( A- r 1 I ) ~ Î¾ (1) = ~ â‡’ ( A- r 1 I ) ~ Î¾ (1) = ~ â‡’ ( A- r 1 I ) ~ Î¾ (1) = ~ â‡’ ( A- r 2 I ) ~ Î¾ (1) = ~ . Thus, two complex vector solutions of ~x = A~x are ~x (1) = ~ Î¾ (1) e r 1 t , ~x (2) = ~ Î¾ (1) e Â¯ r 1 t = ~x (1) . [NOTE: once we know ~x (1) , we know ~x (2) !] By writing ~ Î¾ (1) = ~a + i ~ b (where ~a and ~ b are real vectors), it follows from Eulerâ€™s Formula that ~x (1) ( t ) = ~ Î¾ (1) e r 1 t = ( ~a + i ~ b ) exp { ( Î» + iÎ¼ ) t } = ( ~a + i ~ b ) e Î»t e iÎ¼t = e Î»t ( ~a + i ~ b )( cos Î¼t + i sin Î¼t ) = e Î»t ~a cos Î¼t + i~a sin Î¼t + i ~ b cos Î¼t- ~ b sin Î¼t = e Î»t ~a cos Î¼t- ~ b sin...
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M334Lec32 - Math 334 Lecture#32 Â 7.6 Complex Eigenvalues...

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