If A and C are both nxn matrices that have n distinct different eigenvalues.
AC=CA.
How do you show that the eigenvectors of A and C are the same? How do the eigenvalues relate?
suppose is an eigenvector of so for some scalar let be the eigenspace corresponding to since all eigenvalues of are distinct, is one dimensional, i.e.
now we have thus hence for some scalar that means is also an eigenvector of so we've proved that every eigenvector of is an
eigenvector of a similar argument shows that every eigenvector of is an eigenvector of