given two commuting matrices A and B so that ABBA show that

given two commuting matrices A and B, so that AB=BA, show that if x is an eigenvector of A then it is also an eigenvector of B, but with a possibly different eigenvalue

Solution

Let be an eigenvalue of A, then Ax=x

Similarly if is an eigenvalue of AB, then (AB)x=x ……………..(1)

Now, Multiplying (1) both sides by B

          B(AB)x = Bx

We know that, for three matrices A, B, C

          (AB)C=A(BC)

Appling same formula we can write

          B(AB)x = Bx

       (BA)Bx = Bx

      We can say BA =

So is an eigenvalue of BA & Bx is eigenvector. ….( Bx not 0)