Quiz 3

EGM 4313 Spring 2009 Quiz 3

NAME:_____________________________

Open book/Closed notes/No calculators or computers 30 minute time limit  3 2 4   The matrix A   2 0 2  has the eigenvalue   1. Find the other eigenvalues and as  4 2 3  many linearly independent eigenvectors as possible. Solution: First note that three linearly independent eigenvectors is guaranteed because A is real and symmetric. 3 2 4

2 4 2  2   3  6 2  15  8      1    8   0 2 3 Therefore   1,  1, 8 Eigenvectors for   1:  4 2 4   1   0        2 1 2   2    0  21   2  23  0  4 2 1     0     3   Notice that we can make two arbitrary choices. One way to make linearly independent eigenvectors is to select 1  1,  2  0  3  1 and 1  0,  2  2  3  1 to get the  1  0    eigenvectors  0 ,  2 . Linear combinations of these eigenvectors are also  1  1     eigenvectors, so there are many different ways to find two linearly independent ones. Eigenvector for  5 2 4      2 8 2   4 2 5    

  8: 1   0    2     0  Let 1  2  2 2  43  10,  8 2  23  4   2  1, 3  2   3    0

 2   Therefore  1  is an eigenvector. Any multiple of this is also an eigenvector other than  2   multiplying by 0.