419quiz10.1 | University of Michigan

University:
University of Michigan
Course:
MATH 419 | Linear Spaces and Matrix Theory
Academic year:

2019
Views:

136

Pages:

1
Author:

K Moody

1 −1 1. Let A = . Find all eigenvalues and all eigenvectors of A. 2 4 Solution. To find the eigenvalues, we use the shortcut. The sum of the eigenvalues is the trace of A, that is, 1 + 4 = 5. The product of the eigenvalues is the determinant of A, that is, 1 · 4 − (−1) · 2 = 6, from which the eigenvalues are 2 and 3. Next, we find the eigenvectors. The eigenvectors with the eigenvalue 2 are the non-zero vectors ~x satisfying the equation A~x = 2~x, that is, (A − 2I)~x = ~0, that is,     x1 x2 x1 x2  −1 −1 | 0  −→  1 1 | 0, 2 2 | 0 0 0 | 0 from which the non-zero solutions are −x2 −1 = x2 , for any x2 6= 0. x2 1 Similarly, the eigenvectors with the eigenvalue 3 are the non-zero vectors ~x satisfying the equation A~x = 3~x, that is, (A − 3I)~x = ~0, that is,     x1 x2 x1 x2  −2 −1 | 0  −→  1 1/2 | 0  , 2 1 | 0 0 0 | 0 from which the non-zero solutions are −1/2x2 −1/2 = x2 , x2 1 for any x2 6= 0. Answer. The eigenvaluesare 2 and 3. The eigenvectors with the eigen−1 value 2 are the vectors t , where t can be any non-zero number. 1 −1 The eigenvectors with the eigenvalue 3 are the vectors t , where t 2 can be any non-zero number.

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