Smith Normal Form

A canonical diagonal form for an integer (or more generally PID) matrix achieved by left and right multiplication by unimodular matrices: there exist U, V unimodular over Z such that U A V = diag(d1, d2, ..., dr, 0, ...), where each di divides the next; the diagonal entries are the invariant factors of the associated module.

A canonical diagonal form for matrices over the integers (or more generally over a principal ideal domain) obtained by left and right multiplication by unimodular matrices, whose diagonal entries d1, d2, ... satisfy divisibility d1 | d2 | ... and encode the module structure of the cokernel.

A canonical diagonal form obtainable for a matrix over a principal ideal domain (PID) via invertible row and column operations, producing diagonal entries d1,…,dr (possibly with trailing zeros) satisfying d1 | d2 | … | dr; the diagonal entries are unique up to multiplication by units and encode invariant factors of the corresponding module.