sol61 - Selected Solutions, Leon 6.1 6.1.1 Find the...

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Selected Solutions, Leon § 6.1 6.1.1 Find the eigenvalues and associated eigenspaces of each of the following matrices. (e) A = ± 1 1 - 2 3 ² . The characteristic polynomial is p ( λ ) = λ 2 - 4 λ + 5, with roots λ 1 = 2 - i and = λ 2 = 2 + i . We know that the associated eigenvectors will come in conjugate pairs, so our work is cut in half. We find that A - λ 1 I = ± - 1 + i 1 - 2 1 + i ² , and verify that the matrix is singular (row 2 is (1+ i ) times row 1). If the eigenvector we seek has the form ( x 1 ,x 2 ) T , then setting x 2 = s we have 2 x 1 = (1 + i ) s , or x 1 = ( 1+ i 2 ) s . If we choose s = 2, we have x = (1+ i, 2) T . The associated eigenspace is Span( x ). The eigenspace associated with λ 2 , then, is Span ( (1 - i, 2) T ) . (f) A = 0 1 0 0 0 1 0 0 0 . The characteristic polynomial is p ( λ ) = - λ 3 . Setting p ( λ ) = 0, we find that λ = 0 is an eigenvalue of algebraic multiplicity 3. But N ( A - 0 I ) = N ( A ) = Span ( (1 , 0 , 0) T ) , a 1-dimensional subspace of R 3 . (g) A = 1 1 1 0 2 1 0 0 1 . The characteristic polynomial is p ( λ ) = (1 - λ ) 2 (2 - λ ). Setting p ( λ ) = 0, we find λ 1 = λ 2 = 1 and λ 3 = 2. It remains to be seen whether we can find two linearly independent eigenvectors associated with the repeated eigenvalue. We compute A - I = 0 1 1 0 1 1 0 0 0 . The free variables are
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sol61 - Selected Solutions, Leon 6.1 6.1.1 Find the...

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