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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 $\ge 1$.2012-09-14
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    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 $T$.2012-09-14
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    (Notice that the nilpotency of the matrices plays no role here.)2012-09-14

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