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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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A nullity is a number. They don't have an intersection. I assume you mean the dimension of the intersection of the null spaces of $A$ and $B$ is $\ge 1$. –  Robert Israel Sep 14 '12 at 18:10
    
Pick a basis of the intersection of the two kernels and complete it to a basis of the whole space; use the vectors as columns of $T$. –  Mariano Suárez-Alvarez Sep 14 '12 at 18:10
    
Thank you Mariano –  zacarias Sep 14 '12 at 18:20
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(Notice that the nilpotency of the matrices plays no role here.) –  Mariano Suárez-Alvarez Sep 14 '12 at 18:29
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1 Answer

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. –  zacarias Sep 14 '12 at 18:22
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