= Determinantal divisor
{title2=$\Delta_j(M)=\gcd\{\text{minors of size }j\text{ of }M\}$}
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 $j\leq r$, so $d_j=\Delta_j/\Delta_{j-1}$.
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