Smith normal form

For every matrix $A$ over a principal ideal domain, there are invertible matrices $U,V$ such thatThe nonzero diagonal entries are unique up to multiplication by units. Over the integers, $U$ and $V$ are unimodular and the $d_i$ may be chosen positive; their products $d_1\cdots d_k$ are the greatest common divisors of the $k$-rowed minors.

For an integer matrix $M$, its $j$th determinantal divisor is the nonnegative greatest common divisor of all $j$-by-$j$ matrix minors, with ${\Delta }_0=1$. If all these minors vanish, take ${\Delta }_j=0$. Unimodular row and column operations preserve these divisors. For nonzero invariant factors $d_1\mid \cdots \mid d_r$ in the Smith normal form, ${\Delta }_j=d_1\cdots d_j$ for $j\le r$, so $d_j={\Delta }_j/{\Delta }_{j-1}$.