eigen values and eigen vectors?

Question

I have a matrix that is:
3 2 2
0 1 0
1 3 2

which has eigenvalues 1 (double) and 4.

for the eigenvalue 4, with algebraic multiplicity 1,
its eigenvector is (2 0 1), which implies that the eigenvalue 4 has geometric multiplicity 2, which is higher than its algebraic multiplicity, but i've read that geometric multiplicity is always <= algebraic multiplicity.

what's wrong??

Answer 1

Why on earth do you think the eigenvalue 4 has geometric multiplicity 2?

The eigenspace of this eigenvalue is {t.(2 0 1), t ∈ R} which is a one-dimensional space, so the geometrical multiplicity is 1.

To get geometric multiplicity 2 you need to find two linearly independent eigenvectors for the given eigenvalue.