characterising the square 2x2 matrices on types of eigenvalues

Question

If A is a 2 X 2 matrix with complex entries, then A is similar over $C$ to a matrix of one of the two types :-

1. \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}
2. \begin{bmatrix} a & 0 \\ 1 & a \end{bmatrix} We know we get 2 eigenvalues since the matrix is over complex numbers. If the Eigenvalues are distinct, or if they are equal with geometric multiplicity 2 then it's similar to type 1. If they are equal but the geometric multiplicity is 1 then we get one eigenvector, how do I find out the other basis element?

Answer 1

If $v\neq(0,0)$ is such that $A.v=av$, solve the equation $A.w=aw+v$. That will give you the other vector that you are looking for.