Let $\mathbb K$ be a field and $A, B\in M_n(\mathbb K)$ be nilpotent matrices. Suppose that $nullity(A)\cap nullity(B)\geq 1$.
Can we find a regular matrix $T$ such that the first columns of the two matrices $T^{-1}AT$ and $T^{-1}BT$ are zero?
Let $\mathbb K$ be a field and $A, B\in M_n(\mathbb K)$ be nilpotent matrices. Suppose that $nullity(A)\cap nullity(B)\geq 1$.
Can we find a regular matrix $T$ such that the first columns of the two matrices $T^{-1}AT$ and $T^{-1}BT$ are zero?