What is it known of the behavior of the recurrence $$B_{k+1}=\frac{AB_{k}}{||AB_{k}||}$$ ,i.e, the power iteration recurrence when the vector is replaced by a matrix (if $A$ has dimensions $n\times n$, consider $B$ of dimensions $n\times d, d \geq 2$) and in the denominator we have a matrix norm?
When $d=2$, it can be seen as a (discrete) polygonal flow. I've tried some simulations and the points tend to converge in a line or diverge depending of the eigenvalues of $A$. Note that the recurrence is a bit different from what is employed in the generalized eigenvalue problem: $x_{k+1}= B^{-1}Ax_{k}$, where $x_{k}$ is a vector.