I have been trying to work some problems about determining a set of invariant factors given a relations matrix (in the sense of Jacobson) and vice versa. I am stuck and not sure if I am using the right tools. I have a couple questions but first I will state the relevant definitions and facts from Jacobson Algebra I.

Let $D$ be a principal ideal domain and let $M$ be a module over $D$ generated by a finite set $\{x_1, \ldots , x_n\}$. First let $D^{(n)}$ denote the free module of rank $n$ over $D$ with generators $\{e_1, \ldots, e_n \}$ and consider the epimorphism $ \eta : \sum_{1}^{n} a_i e_i \rightarrow \sum_{1}^{n} a_i x_i$, $a_i \in D$. It follows that $ M \cong D^{(n)}/ K$ where $K = \ker \eta$ and by theorem 3.7 of Jacobson $K$ has a base of $m \leq n$ elements denoted by $\{ f_1 , \ldots , f_m \}$ . We then express the generators of $K$ in terms of the base $\{e_1, \ldots, e_n \}$ in the form

$$\begin{align*} f_1 &= a_{11} e_1 + a_{12} e_2 + \cdots + a_{1 n}e_n \\ f_2 &= a_{21} e_1 + a_{22}e_2 + \cdots + a_{2 n}e_n \\ \vdots \\ f_m &= a_{m1}e_1 + a_{m 2 } e_2 + \cdots + a_{mn} e_n \end{align*}$$

Then the $m \times n$ matrix $A = (a_{ij})$ of these relations is called the relations matrix for the ordered set of generators $\{ f_1 , \ldots , f_m \}$ in terms of the ordered basis $\{e_1, \ldots, e_n \}$.

Fundamental Theorem: Let $D$ be a pid and let $M$ be a finitely generated $D$ module. Then $M$ is isomorphic to the direct sum of finitely many cyclic modules. More precisely, $M \cong D^r \oplus D/(a_1) \oplus D/(a_2) \oplus \ldots \oplus D/(a_m)$ for some integer $ r \geq 0$ and nonzero elements $a_1, a_2, \ldots, a_m$ called the invariant factors which are not units in $D$ and satisfy the divisibility relations $a_1 | a_2 | \ldots | a_m$.

I am trying to use the relations matrix to study the direct sum decomposition of $M$ in terms of the fundamental theorem for finitely generated modules over a p.i.d. I am trying to determine the quickest method for computing invariant factors given a relations matrix (because I will eventually take a qualifying exam on many of these type of problems). So far the only tool I have found in this regard is the following:

Jacobson Theorem 3.9: Let $A$ be an $m \times n$ matrix with entries in a p.i.d $D$ and suppose the rank of $A$ to be $r$. For each $i \leq r$ let $\Delta_i$ be a g.c.d of the $i$-rowed minors of $A$. Then any set of invariant factors for $A$ differ by unit multipliers from the elements $d_1 = \Delta_1, d_2 = \Delta_2 \Delta_{1}^{-1}, \ldots , d_r = \Delta_r \Delta_{r-1}^{-1}$.

Question 1: What is meant by $i$-th rowed minor in this context? The definition I am familiar with is for a minor usually involves $M_{i,j}$ specifying which row and column is "crossed out" before we take a determinant.

Question 2: Let $V$ be a finite-dimensional vector space over the field $F$ and let $T: V \rightarrow V$ be a linear operator. Give $V$ an $F[x]$ module structure by defining $x \alpha = T(\alpha)$ for each $\alpha \in V$. If $F = \mathbb{C}$ and let $$ A = \left( \begin{array}{ccc} x^2(x-1)^2 & 0 & 0 \\ 0 & x(x-1)(x-2)^2 & 0 \\ 0 & 0 & x(x-2)^3 \end{array} \right) $$ be a relation matrix for V with respect to $\{v_1, v_2, v_3\}$ generators of $V$. How do we use theorem 3.9 to determine the invariant factors of A with respect to $\{v_1, v_2, v_3\}$ or is the a better way than the theorem I listed? (I have struggled to find any examples in this sort of language I am using so any reference suggestions would be greatly appreciated as well).

Question 3: Given a set of invariant factors is it possible to determine a relations matrix corresponding to the invariant factors? Or how to we go to from a normal form to a relations matrix.

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    $\begingroup$ Do your books cover an algorithm leading to a Smith normal form of a matrix (over a Euclidean domain or a PID)? That gives an algorithmic way of finding the invariant factors as well as the related module bases. In your example cases there are relatively few interesting minors, so using the Theorem is quite straightforward, if you only need to find the invariant factors and don't need to find the stacked bases. $\endgroup$ – Jyrki Lahtonen Jul 21 '11 at 14:21
  • $\begingroup$ yes there is such an algorithm and it definitely applies in this case. After re-reading the proofs I realize I am having some fundamental problems understanding a couple definitions in particular I added a new question 1 about notation for minors. $\endgroup$ – user7980 Jul 22 '11 at 3:38
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    $\begingroup$ You get an $i$-rowed minor by picking any $i$ (distinct) rows and any $i$ columns, and computing the determinant of the thus created $i\times i$ matrix. In general there are ${n\choose i}$ ways of selecting the rows and ${n\choose i}$ way of selecting the columns, so ${n\choose i}^2\,$ $i$-rowed minors, so that's a lot of minors. In your exam cases the problem does appear to be manageable as there are many zeros. $\endgroup$ – Jyrki Lahtonen Jul 22 '11 at 16:30

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