## Do commutative matrices share the same eigenvectors?

Let two square matrices A and B represent linear operators on a vector space V over C. Suppose they are commutative. Then ABx=BAx,∀x∈V Then let ˜x be an eigenvector of B. Setting, x=˜x, we see that AB˜x=BA˜xAλ˜x=BA˜x,∃λ∈Cλ(A˜x)=B(A˜x)⟹A˜x=α˜x,∃α∈C So every eigenvector of B is also an eigenvector of A. ◼ MY QUESTIONS: Is this valid? If … Read more