Smith normal form theorem

Smith normal form theorem: Let $R$ be a PID and let $A$ be an $m\times n$ matrix with entries in $R$. Then there exist invertible matrices $U\in \mathrm{GL}_m(R)$ and $V\in \mathrm{GL}_n(R)$ such that

where $d_i\neq 0$ for $1\le i\le r$ and $d_1\mid d_2\mid \cdots \mid d_r$. The $d_i$ are determined uniquely up to multiplication by units; they are the Smith invariants (see Smith normal form invariants ).

Interpreting $A$ as the matrix of a homomorphism between free modules (compare matrix representation ), Smith normal form yields the invariant factor decomposition in the structure theorem for finitely generated modules over a PID by identifying the cokernel as a direct sum of cyclic modules $R/(d_i)$.

Proof sketch: Use elementary row/column operations corresponding to multiplication by invertible matrices to perform a Euclidean-algorithm-style reduction on entries, producing a diagonal form with each diagonal entry dividing the next. The PID property ensures principal generators for ideals arising during the reduction.