How do I find the eigenvectors for a 3x3 matrix

I have the following matrix A:
5 2 -3
4 5 -4
6 4 -4

I have never done eigenvectors before but I was told that the eigenvalues are 1, 2 and 3. Could anyone show me the detailed solution for this problem please?

λ is an eigenvalue of A iff A ν = λ ν for some non-zero vector ν, or
A ν − λ ν = 0
A ν − λ I ν = 0
(A − λ I) ν = 0

Since vector ν is a non-zero vector, then (A − λ I) ν = 0 iff
determinant of matrix (A − λ I) = 0
[ 5−λ .... 2 .... −3 . ]
[ . 4 .... 5−λ .. −4 . ]
[ . 6 ...... 4 .. −4−λ ]
= 0

(5−λ) ((5−λ)(−4−λ) + 16) − 2 (4(−4−λ) + 24) − 3 (16−6(5−λ)) = 0
−λ³ + 6λ² + 15λ − 100 + 80 − 16λ + 32 + 8λ − 48 − 48 + 90 − 18λ = 0
−λ³ + 6λ² − 11λ + 6 = 0
−(λ − 1) (λ − 2) (λ − 3) = 0
λ = 1, λ = 2, λ = 3

Now you need to find eigenvectors ν such that for each λ : A ν = λ ν

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So we go back to matrix (A − λ I) with λ = 1, then reduce it:
[ 4 2 −3 ] ...... [ 4 2 −3 ] ..... [ 4 0 −2] ...... [ 2 0 −1] [ν₁] .. [0]
[ 4 4 −4 ] ---> [ 0 2 −1 ] ---> [ 0 2 −1] ---> [ 0 2 −1] [ν₂] = [0]
[ 6 4 −5 ] ...... [ 0 4 −2 ] ..... [ 0 0 0 ] ....... [ 0 0 0 ] [ν₃] ... [0]

If we let ν₁ = t, then
2ν₁ − ν₃ = 0 -----> ν₃ = 2ν₁= 2t
2ν₂ − ν₃ = 0 -----> ν₂ = ν₃/2= t

So eigenvectors for λ = 1 are [ t t 2t ]ᵀ = t [ 1 1 2 ]ᵀ

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Now we go back to matrix (A − λ I) with λ = 2, then reduce it:
[ 3 2 −3 ] ...... [ 3 2 −3] .... [ 3 0 −3] ...... [ 1 0 −1] [ν₁] .. [0]
[ 4 3 −4 ] ---> [ 0 1 0 ] ---> [ 0 1 0 ] ----> [ 0 1 0 ] [ν₂] = [0]
[ 6 4 −6 ] ...... [ 0 0 0 ] ..... [ 0 0 0 ] ....... [ 0 0 0 ] [ν₃] ... [0]

Let ν₁ = t, then
ν₂ = 0
ν₁ − ν₃ = 0 -----> ν₃ = ν₁= t

So eigenvectors for λ = 2 are [ t 0 t ]ᵀ = t [ 1 0 1 ]ᵀ

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We go back to matrix (A − λ I) again with λ = 3, then reduce it:
[ 2 2 −3 ] ...... [ 2 2 −3] ..... [ 2 0 −1] [ν₁] .. [0]
[ 4 2 −4 ] ---> [ 0 2 −2] ---> [ 0 1 −1] [ν₂] = [0]
[ 6 4 −7 ] ...... [ 0 2 −2] ..... [ 0 0 0 ] [ν₃] ... [0]

Let ν₁ = t, then
2ν₁ − ν₃ = 0 -----> ν₃ = 2ν₁= 2t
ν₂ − ν₃ = 0 -----> ν₂ = ν₃ = 2t

So eigenvectors for λ = 3 are [ t 2t 2t ]ᵀ = t [ 1 2 2 ]ᵀ

Go to:
http://www.wolframalpha.com

In the input box you can type: eig[{{5, 2, -3},{4, 5, -4},{6, 4, -4}}]

This will give you your answers.

How do you actually do this by hand? It is fairly laborious, this video gives a good talk on how you can do it. When I have eigenvalue problems, I always use the computer.

http://www.youtube.com/watch?v=11dNghWC4…