if matrix A and matrix B are commute
then prove that A & B have the same eigenvectors and vice reverse
plz replay before 12 am today
I think some assumptions are missing.
do commute, they both have only one eingevalue 1, but the eigenspace of B is only one-dimensional, whereas every vector is eigenvector of A.
However, you can show:
If AB=BA and A has n distinct eigenvalues, then B has the same eigenvectors as A.
First, try to show the following easy lemma: If D is a diagonal matrix with distinct diagonal elements and DX=XD, then X is diagonal as well.
Now, you have
for some diagonal matrix. The columns of P are precisely the eigenvectors of A. From
Now, using the above lemma, the matrix
is diagonal. Let us denote it
which implies that B has the same eigenvectors.