5.5-lay - 1 5.5 Complex Eigenvalues Homework: 2, 6, 8, 10,...

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5.5 Complex Eigenvalues Homework : 2, 6, 8, 10, 14, 18 Goal : To Uncover “hidden’ information of certain matrices with real entries with complex eigenvalues. Ex 1 : Let A be the matrix of a rotation through 90°. Then A= ° ° ° - ° 90 cos 90 sin 90 sin 90 cos = - 0 1 1 0 det (A – λI) = If λ = i , then A – λI = If λ = - i , then A – λI = Ex 2 : Let A = - 1 . 1 75 . 0 6 . 0 5 . 0 . det (A – λI) = det - - - λ 1 . 1 75 . 0 6 . 0 5 . 0 = 1 6 . 1 2 + - λ = 0.8 i 6 . 0 ± If λ = 0.8 + i 6 . 0 , then A – λI = 1
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λ = 0.8 - i 6 . 0 , then v 2 = - + - = - - 0 4 5 2 5 4 2 i i Graph of A x i A = - 1 . 1 75 . 0 6 . 0 5 . 0 and A = - 1 . 1 75 . 0 6 . 0 5 . 0 x 0 = 0 2 . x 1 = A x 0 = 5 . 1 1 x 2 =A x 1 =A(A x 0 ) = A 2 x 0 = - 4 . 2 4 . , Real and Imaginary Parts of Vectors
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5.5-lay - 1 5.5 Complex Eigenvalues Homework: 2, 6, 8, 10,...

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