Find the eigenvectors of the matrix
I've already found the eigenvalues which come out to be 1, 2+i and 2-i.
To find the eigenvectors I have to find v such that .
I already found the eigenvector for . I'm having trouble with it for the complex cases.
Here's my working for :
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The only vector that satisfies these conditions is the zero matrix which cannot be an eigenvector by definition!
By the way, are you working with a vector space over the real numbers or the complex numbers. Since the matrix you give has only real entries, it could be either one. But if you are working with a vector space over the real numbers, then only "1" is an eigenvalue.
The question itself doesn't say, but i'm guessing it's over the complex numbers.
Also, I think there's something wrong with my above working. Checking on octave gives:
A =
-3 17 -4
-2 9 -2
-2 8 -1
P =
1.00000 - 0.00000i
0.38462 + 0.07692i
0.00000 - 0.30769i
octave-3.0.1:4> B=A*P
B =
3.5385 + 2.5385i
1.4615 + 1.3077i
1.0769 + 0.9231i
octave-3.0.1:6> B./P
ans =
3.5385 + 2.5385i
4.3077 + 2.5385i
-3.0000 + 3.5000i
But B./P should be an eigenvector, either 2-i or 2+i.
Furthermore, actually computing the answer on matlab gives:
A =
-3 17 -4
-2 9 -2
-2 8 -1
octave-3.0.1:2> [V,D] = eig(A)
V =
0.87447 + 0.00000i 0.87447 - 0.00000i -0.70711 + 0.00000i
0.33634 + 0.06727i 0.33634 - 0.06727i -0.00000 + 0.00000i
0.33634 + 0.06727i 0.33634 - 0.06727i 0.70711 + 0.00000i
D =
2.00000 + 1.00000i 0.00000 + 0.00000i 0.00000 + 0.00000i
0.00000 + 0.00000i 2.00000 - 1.00000i 0.00000 + 0.00000i
0.00000 + 0.00000i 0.00000 + 0.00000i 1.00000 + 0.00000i
...and this is clearly different to what I have.
From what I can see, my initial method to find the eigenvectors should have worked!