Homework SourseFind all diagonal 3 times 3 matrices A with complex entries
Find all diagonal 3 times 3 matrices A with complex entries such that A^2 = [0 0 0 0 2 0 0 0 -1]
![Find all diagonal 3 times 3 matrices A with complex entries such that A^2 = [0 0 0 0 2 0 0 0 -1]SolutionThe characteristic equation of A2 is det (A2- I3) = 0 o](/WebImages/38/find-all-diagonal-3-times-3-matrices-a-with-complex-entries-1116321-1761593043-0.webp "Find all diagonal 3 times 3 matrices A with complex entries such that A^2 = [0 0 0 0 2 0 0 0 -1]SolutionThe characteristic equation of A2 is det (A2- I3) = 0 o")
![Find all diagonal 3 times 3 matrices A with complex entries such that A^2 = [0 0 0 0 2 0 0 0 -1]SolutionThe characteristic equation of A2 is det (A2- I3) = 0 o](/WebImages/38/find-all-diagonal-3-times-3-matrices-a-with-complex-entries-1116321-1761593043-1.webp "Find all diagonal 3 times 3 matrices A with complex entries such that A^2 = [0 0 0 0 2 0 0 0 -1]SolutionThe characteristic equation of A2 is det (A2- I3) = 0 o")
Solution
The characteristic equation of A2 is det (A2- I3) = 0 or, (-2)( +1) = 0. Therefore, the eigenvalues of A2 are 1 = 2, 2 = -1 and 3 = 0. The eigenvectors of A2 corresponding to the eigenvalue are solutions to the equation (A2- I3)X = 0. These are v1 = ( 0,1,0)T, v2 = ( 0,0,1)T and v3= (1,0,0)T. Then A2 = PDP-1 where P =
0
0
1
1
0
0
0
1
0
P-1 =
0
1
0
0
0
1
1
0
0
and D =
2
0
0
0
-1
0
0
0
0
Finally, A = PD1/2P-1. Here, D1/2 =
±2
0
0
0
±i
0
0
0
0
Therefore A =
0
0
0
0
±2
0
0
0
±i
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
