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.