2

Let $A$ be an $n\times n$ matrix over a field $F$. Denote by the same letter $A$ the linear operator $F^n\to F^n$ given by $X\mapsto AX$. Endow $F^n$ with the structure of an $F[t]$-module by defining scalar multiplication as follows: if $f(t)\in F[t]$ and $X\in F^n$, then $f(t)X=[f(A)]X$. By the structure theorem for modules over PIDs, $F^n$ is a direct sum of cyclic modules each of which is of the form $F[t]/(g(t))$ where $g(t)$ is a power of a monic irreducible polynomial.

The question is how to find this decomposition in practice? I believe one should reduce some matrix to the Smith normal form and then draw conclusions. But I can't seem to adapt the usual alorithm for groups for this case. I guess it would be best if someone could give some (nontrivial) example (say with $n=3$, $F=\mathbb R$, and $A$ a matrix of your choice).

user557
  • 11,889

1 Answers1

1

The key result is that we have an isomorphism of $F[X]$-modules $F^n\simeq F[X]^n/\ker(XI_n-A)$.

I don't have much time now, so I leave you to find some references for now on the web or in the standard books. If I have time tonight (French time) , I will edit my answer and provide a full proof.

Now, you just apply the standard procedure . Find the Smith normal form of $XI_n-A$: $$\begin{pmatrix} I_{n-r} & & & \cr & P_1 & & \cr & & \ddots & \cr & & & P_r\end{pmatrix},$$ where $P_1,\ldots, P_r\in F[X]$ are monic of degree $\geq 1$ and $P_1\mid P_2\mid\cdots\mid P_r.$

Then $F^n\simeq F[X]/(P_1)\times\cdots \times F[X]/(P_r)$.

GreginGre
  • 15,028
  • Thanks. I guess the main fact you use is one of the exercises in Artin's text which I asked a separate question about: https://math.stackexchange.com/questions/2725494/finding-a-presentation-matrix-for-an-ft-module – user557 Apr 07 '18 at 03:12
  • Ok. I will write a detailed solution on monday when i'm back home then. – GreginGre Apr 07 '18 at 09:18