Fundamentals of Matrix Algebra, 2011a

4.2 Properes of Eigenvalues and Eigenvectors
2. We first compute the inverses of A and B. They are:
⎡
⎤
[ ]
−4 1/3 13/3
A −1 −1/8 5/24
=
and B −1 = ⎣ −3/2 1/2 3/2 ⎦ .
1/24 1/24
−3 1/3 10/3
Finding the eigenvalues and eigenvectors of these matrices is not terribly hard,
but it is not “easy,” either. Therefore, we omit showing the intermediate steps
and go right to the conclusions.
For A −1 , we have eigenvalues λ = −1/6 and 1/12, with eigenvectors
[ ] [ ]
−5
1
⃗x = x 2 and x
1
2 , respecvely.
1
For B −1 , we have eigenvalues λ = −1, 1/2 and 1/3 with eigenvectors
⎡
⃗x = x 3
⎣ 3 ⎤ ⎡
1 ⎦ , x 3
⎣ 2 ⎤ ⎡
1 ⎦ and x 3
⎣ 1 ⎤
0 ⎦ , respecvely.
2 2
1
3. Of course, compung the transpose of A and B is easy; compung their eigenvalues
and eigenvectors takes more work. Again, we omit the intermediate steps.
For A T , we have eigenvalues λ = −6 and 12 with eigenvectors
[ ] [ ]
−1
5
⃗x = x 2 and x
1
2 , respecvely.
1
For B T , we have eigenvalues λ = −1, 2 and 3 with eigenvectors
⎡
⃗x = x 3
⎣ −1 ⎤ ⎡
0 ⎦ , x 3
⎣ −1 ⎤ ⎡
1 ⎦ and x 3
⎣ 0 ⎤
−2 ⎦ , respecvely.
1 1
1
4. The trace of A is 6; the trace of B is 4.
5. The determinant of A is −72; the determinant of B is −6.
.
Now that we have completed the “grunt work,” let’s analyze the results of the previous
example. We are looking for any paerns or relaonships that we can find.
The eigenvalues and eigenvectors of A and A −1 .
In our example, we found that the eigenvalues of A are −6 and 12; the eigenvalues
of A −1 are −1/6 and 1/12. Also, the eigenvalues of B are −1, 2 and 3, whereas the
179