The matrix A 0 0 0 5 5 0 5 5 0 has two real eigenvalues one

The matrix A = [0 0 0 -5 5 0 5 -5 0] has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigenspace. lambda_1 = has multiplicity 1, with a basis of lambda_2 = has multiplicity 2, with a basis of

Solution

For the 0 eigenvalue we are solving

Ax=0

LEt, x=[a b c]

So, Ax=0 gives

-5a+5b=0 ie a=b

So,

x=[a a c]=a[1 1 0]+c[0 0 1]

So basis is

(1,1,0),(0,0,1)