- #1
aznkid310
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[SOLVED] diagonalization, eigenvectors, eigenvalues
Find a nonsingular matrix P such that (P^-1)*A*P is diagonal
| 1 2 3 |
| 0 1 0 |
| 2 1 2 |
I am at a loss on how to do this. I've tried finding the eigen values but its getting me nowhere
i row reduced to 1 2 3
0 1 0
0 0 -4
and found : (x-1)^2 and (x+4), which gives me eigen values of 1 and -4.
Using A - Ix, when x = -4, the matrix becomes 5 2 3 0
0 5 0 0
2 1 6 0
Using row reduction: 1 2/5 1/5 0
0 5 0 0
2 1 6 0
1 2/5 1/5 0
0 1 0 0
0 1/5 28/5 0
1 2/5 1/5 0
0 1 0 0
0 0 28/5 0
1 0 0 0
0 1 0 0
0 0 1 0
which means X = y = z = 0, but that's wrong
When i use x = 1: 0 2 3 0
0 0 0 0
2 1 1 0
this gives 2y = -3z
2X + y + z = 0 ==> 2X = z/2
If z = 4, then x = 1, y = -6
That's one correct answer, but i can't get the second one, which is suppose to be
X = -3, y = 0, z = 2
Homework Statement
Find a nonsingular matrix P such that (P^-1)*A*P is diagonal
| 1 2 3 |
| 0 1 0 |
| 2 1 2 |
Homework Equations
I am at a loss on how to do this. I've tried finding the eigen values but its getting me nowhere
The Attempt at a Solution
i row reduced to 1 2 3
0 1 0
0 0 -4
and found : (x-1)^2 and (x+4), which gives me eigen values of 1 and -4.
Using A - Ix, when x = -4, the matrix becomes 5 2 3 0
0 5 0 0
2 1 6 0
Using row reduction: 1 2/5 1/5 0
0 5 0 0
2 1 6 0
1 2/5 1/5 0
0 1 0 0
0 1/5 28/5 0
1 2/5 1/5 0
0 1 0 0
0 0 28/5 0
1 0 0 0
0 1 0 0
0 0 1 0
which means X = y = z = 0, but that's wrong
When i use x = 1: 0 2 3 0
0 0 0 0
2 1 1 0
this gives 2y = -3z
2X + y + z = 0 ==> 2X = z/2
If z = 4, then x = 1, y = -6
That's one correct answer, but i can't get the second one, which is suppose to be
X = -3, y = 0, z = 2