$ \mathrel y \in C^n and y\neq 0 $ and m be the smallest integer such that $ \{y,By,...,B^my\} $ is a dependent set. How to prove that $ V = span \{y,By,...,B^{m-1}y\} $ is B−invariant.
I came across the above question in my previous year's exam paper. Please let me know how to solve it as I am getting no idea how to approach. My intuition is saying that it should be related to eigenvectors and eigen values.
Since $m$ is the smallest such integer, the vectors $y,By,\dots,B^{m-1}y$ are linearly independent. Adding $B^my$ makes it dependent, so $B^my\in{\rm span}(y,By,\dots,B^{m-1}y)=:W$, otherwise they would be still independent.
But then, $W={\rm span}(y,By,\dots,B^my)$ and for any element $v\in W$ we have $v=\lambda_1y+\lambda_2By+\dots+\lambda_mB^{m-1}y$, so $$Bv=\lambda_1 By+\lambda_2 B^2y+\dots+\lambda_mB^my\ \in W\,.$$