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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?

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    (Notice that the nilpotency o$f$ the matrices plays no role here.)2012-09-14

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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 $b$.

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    Thank you Robert. Yes I mean the dimension of the intersection of the null spaces of $A$ and $B$ is $\geq $ 1.2012-09-14