Consider two n times n matrices C and D that commute under m

Consider two n times n matrices C and D that commute under matrix multiplication, so CD = DC. If x R^n is an eigenvector of C, show that Dx is also an eigenvector of C.

Solution

Let be the eigenvalue of C corresponding to the eigenvector x. Then Cx = x so that (DC)x = D(Cx) = D(x) = (Dx). Since C and D commute, hence C(Dx) = (CD)x = (DC) x =(Dx). This shows that Dx is also an eigenvector of C.