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?
The first column of $T^{-1} A T$ is $T^{-1} A T e_1$ where $e_1 = (1,0,\ldots,0)^T$. So we want $T e_1 = b$ where $Ab = Bb = 0$ and $b \ne 0$. That's the first column of $T$. For the other columns, take any basis of ${\mathbb K}^n$ whose first element is $b$.