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$Let\ A ∈ M_n(F)$\ such that $A^n = 0$ but $A^{n−1} \neq 0.$

$If\ B ∈ M_n(F)$ such that $AB = BA$, prove that $B = a_0 + a_1A + a_2A^2 + ··· +a_{n−1}A^{n−1} $

for some $a_0,...,a_{n−1} ∈ F.$

I only know that $AB^n = B^nA$, and $\{v,Av,...,A^{n-1}v\}$ is a basis of $F^n$,

but I'm not sure if they are useful.

3SAT
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Claire
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  • See also here. – Dietrich Burde Nov 19 '15 at 15:19
  • @DietrichBurde While the linked questions are quite similar, the specific question here allows for a much more elementary answer. – daw Nov 19 '15 at 15:39
  • @daw: Yes, but which has also been given here (there are several solutions for this problem on MSE). – Dietrich Burde Nov 19 '15 at 15:42
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    I found the accepted answer here more appropriate for the diffuclty level of the question: http://math.stackexchange.com/questions/178604/complex-matrix-that-commutes-with-another-complex-matrix/ – daw Nov 19 '15 at 15:46

2 Answers2

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Hint

If $B=\sum_i a_i A^i$ then surely $Bv=\sum_i a_i A^iv$. However the collection $\{A^i_v\}_i$ forms a basis so there is exactly one way to express $Bv$ in this basis. So this gives us a way to find the $a_i$. From here just apply $B$ to all the base vectors.

Jorik
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First, $A^n=0$ and $A^{n-1}\ne0$ means that $A$ is nilpotent, and its characteristic polynomial is equal the minimal polynomial. Thus there is an invertible matrix $S$ such that $$ S^{-1}AS=\pmatrix{0&1&0 \dots\\ 0&0&1&\dots\\ \dots}=:J $$ i.e., it is just one Jordan block to the eigenvalue $0$. Now write $C:=S^{-1}BS$, then it holds $JC=CJ$. Write $C=(c_{ij})$ the we see $$ CJ = \pmatrix{ 0&c_{11}& \dots &c_{1,n-1}\\\vdots &\vdots&& \vdots\\0&c_{n1}& \dots &c_{n,n-1}} = JC=\pmatrix{ c_{21}& \dots &c_{2n}\\\vdots && \vdots\\c_{n1}& \dots &c_{nn}\\ 0 & \dots &0}. $$ We immediately see that $c_{n1}\dots c_{n,n-1}$ and $c_{21}\dots c_{n1}$ must be zero. Moreover $c_{ij}=c_{i+1,j+1}$ for all $i,j=1\dots n-1$. Hence the matrix $C$ has the form $$ C=\pmatrix{ c_{11} & c_{12} & \dots & c_{1n} \\ 0&c_{11} & \dots & c_{1n} \\ \vdots & \ddots &\ddots&\ddots\\ 0&0&\dots&c_{11}}, $$ which is nothing else than $C=\sum_{i=0}^{n-1} c_{1,i+1} J^{i}$. This implies $$ B = SCS^{-1} = \sum_{i=0}^{n-1} c_{1,i+1}S J^{i}S^{-1} =\sum_{i=0}^{n-1} c_{1,i+1}A^i, $$ which is the claim.

daw
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