The invariant factors of a matrix over a principal ideal domain are the nonzero entries 
, ..., 
on the diagonal of its Smith
normal form, ordered so that 
divides 
. They are unique up to multiplication by units.
For an integer matrix, they are conventionally
chosen to be positive.
For a square matrix 
over a field 
, its polynomial invariant
factors are the nonconstant monic polynomials
on the diagonal of the Smith
normal form of 
over ![F[x]](/images/equations/InvariantFactor/Inline8.svg)
, where 
is the identity matrix.
Their companion matrices are the blocks in the
rational canonical form of 
.
