Two matrices such that AB=B and BA=A yield collinear projections?

Question

$A$ and $B$ are two matrices such that $AB=B$ and $BA=A$. For vector $z$, let $x=Az$ and $y=Bz$. Can we show two vectors $x$ and $y$ are in the same direction? Is there any special condition required to yield this conclusion, such as $A$ and $B$ must be projection matrices?

Answer 1

You want that $A,B$ have a common eigenvector.

It suffices that $AB=B$ OR $BA=A$.

For example,assume that $AB=B$, that is, $(A-I)B=0$. Then $A-I,B$ have a common eigenvector and, then, $A,B$ have a common eigenvector.