Math 104 :: Suggested exercise: commuting matrices Problem: Let A and B be two square matrices such that AB = BA (we say that A and B commute). Assume each eigenvalue of A has multiplicity one, i.e., corresponds to a single eigenvector. Then show that B has the same eigenvectors as A. Solution: Write Av = λv. Then A(Bv) = BAv = Bλv = λ(Bv). This says that Bv is also an eigenvector of A with eigenvalue λ. Since this eigenvector is unique up to a scalar, by assumption, Bv must be a scalar multiple of v. So there must exist µ such that Bv = µv. So v is also an eigenvector of B.