# Two problems about Structure Theorem for finitely generated modules over PIDs

1) Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either

$$\begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&1&0&-1\\ 0&0&1&0 \end{pmatrix}, \begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{pmatrix}.$$

2) Let $A= \begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9 \end{pmatrix}$. Find an integer $r>0$ and positive integers $1<d_1\mid d_2\mid\cdots\mid d_s$ such that $\mathbb{Z}^3/A\mathbb{Z}^3$ is isomorphic to $\mathbb{Z}/d_1\mathbb{Z} \times \cdots \times \mathbb{Z}/d_s\mathbb{Z} \times \mathbb{Z}^r$.

I think both problems have something to do with the Structure Theorem for finitely generated modules over PIDs, but I am not familiar with that, would someone please help. Actually I don't even understand what is meant by $\mathbb{Z}^3/A\mathbb{Z}^3$...

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1) The matrix $A$ has the characteristic polynomial $(X^2+1)^2$. Then its minimal polynomial can be $X^2+1$ or $(X^2+1)^2$.
If the minimal polynomial of $A$ is $X^2+1$, then its characteristic matrix is equivalent to the canonical diagonal matrix having $1,1,X^2+1,X^2+1$ on the main diagonal. This corresponds to the decomposition of the $\mathbb R[X]$-module $\mathbb R^4$ given by $$\mathbb R[X]/(X^2+1)\oplus\mathbb R[X]/(X^2+1).$$ In this case your matrix is similar to \begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{pmatrix} which is a matrix made by two (identical) companion matrices associated to the polynomial $X^2+1.$
On the other side, if the minimal polynomial of $A$ is $(X^2+1)^2$, then its characteristic matrix is equivalent to the canonical diagonal matrix having $1,1,1,(X^2+1)^2$ on the main diagonal. This corresponds to the decomposition of the $\mathbb R[X]$-module $\mathbb R^4$ given by $\mathbb R[X]/((X^2+1)^2)$. In this case your matrix is similar to \begin{pmatrix} 0&0&0&-1\\ 1&0&0&0\\ 0&1&0&-2\\ 0&0&1&0 \end{pmatrix} which is the companion matrix associated to the polynomial $(X^2+1)^2$.
2) I don't know if the first question has something to do with the structure theorem of modules over a PID, but the second definitely has via the Smith Normal Form. Here $A\mathbb Z^3$ denotes the $\mathbb Z$-submodule of $\mathbb Z^3$ generated by the rows (or columns) of $A$ and coincides to the $\mathbb Z$-submodule of $\mathbb Z^3$ generated by the rows (or columns) of its SNF. Since the SNF of $A$ is \begin{pmatrix} 1&0&0\\ 0&3&0\\ 0&0&0 \end{pmatrix} we get that $\mathbb{Z}^3/A\mathbb{Z}^3$ is isomorphic to $\mathbb Z/3\mathbb Z\oplus\mathbb Z$.
thanks YACP, do you mind explain more about the first problem...? I still don't quite get it. My professor gives me some hints: The characteristic polynomial of $A$ is $f_A(x)=(x^2+1)^2.$ Therefore, by structure theorem of modules over a PID, $Im (A)$ is isomorphic to $\mathbb{R}[x]/((x^2+1)^2)$ or $\mathbb{R}[x]/(x^2+1) \times \mathbb{R}[x]/(x^2+1)$... – Ishigami May 18 '13 at 12:35