Let A be an n×n matrix over a field F. Using the three elementary row and column operations over elements in the field, the n×n matrix x I - A with entries from the principal ideal domain F[x] (where I is the identity matrix) can be put into the diagonal form [1 | 0 | ... | 0 | 0 | 0 | 0 | 0 0 | 1 | ⋱ | 0 | 0 | 0 | 0 | 0 ⋮ | ⋱ | ⋱ | ⋱ | ⋱ | ⋱ | ⋱ | ⋮ 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 0 | 0 | 0 | 0 | a_1(x) | 0 | 0 | 0 0 | 0 | 0 | 0 | 0 | a_2(x) | 0 | 0 ⋮ | ⋱ | ⋱ | ⋱ | ⋱ | ⋱ | ⋱ | ⋮ 0 | 0 | 0 | 0 | 0 | 0 | 0 | a_m(x)],

Hermite normal form | normal form