Relation of eigenvectors and eigenvalues between commuting matrices

Question

Let $A,B\in\mathbb{C}^{n\times n}$ commute, let $B$ be nonsingular and let $\lambda\in\sigma(A)$ have geometric multiplicity 1 and a corresponding eigenvector $\mathbf{v}.$

I have so far shown that $B\mathbf{v}$ is an eigenvector of $A$ corresponding to $\lambda$ and I have to show that

1. There exists a $\mu\in\sigma(B)$ such that $\mathbf{v}$ is an eigenvector of $B$ corresponding to $\mu.$
2. If $B=B^{-1},$ then $\mu=\pm 1.$

I am a bit stuck at both of these parts in how to tackle them, I think part 1 requires use of the multiplicity and that the eigenspace will be one dimensional but I am not sure.

Answer 1

1. Because the geometric multiplicity of $\lambda$ is $1$, the eigenvectors $v$ and $Bv$ must be parallel.

2. Assuming $B=B^{-1}$, we have $v=B^{-1}Bv=B^2v=\mu^2v$, hence $\mu^2=1$.