Invariant Factor

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 , where is the identity matrix. Their companion matrices are the blocks in the rational canonical form of .