I want to find the solution by characteristic value of:

1 -1 -1
0 1 3
0 3 1


I understand how to get the eigen values of lamba=1, 4, -2

If we plug lamba =4 we get:

-3 -1 -1 k1 = 0
0 -3 3 k2 = 0
0 3 -3 k3 = 0

I understand up to this point. What I don't understand is how we get a non-zero solution is the characteristic vector:

2
-3
3

from which we get

2exp(4t)
-3exp(4t)
-3exp(4t)


I understand the exponent part with the 4t, I just don't understand how we get the characteristic vector and then the solution from it. Could anyone explain? Thanks,

Kim