Consider a $2\times2$ matrix $P$ with entries in $\mathbb{Z}$ and $\det(P)=N$. Its (row-wise, lower) Hermite Normal Form is given by
$$
H=\begin{pmatrix} d & 0 \\ s & \bar{d}\equiv N/d\end{pmatrix},\ 0 \le s < d.
$$
Let the Smith decomposition of $P$ be given by $P=URV$, where
$$
R=\begin{pmatrix} c_1 & 0 \\ 0 & c_2\end{pmatrix},c_1\mid c_2,\ c_2=N/c_1.
$$
I have observed the following (and checked using brute force for all cases with $N \le 1000$) that $c_1=\mathrm{GCD}\left(d,s,\bar{d}\right)$.
My questions are: (i) is this really the case, and if so (ii) how can I prove this.
Best Answer
Yes, this is really the case. If two matrices are equivalent they have the same elementary divisors, so the matrices $H$ and $R$, which are both equivalent to $P$, have $\gcd(d,s,\overline{d})=\gcd(c_1,c_2)$.