1.
Describe a canonical representation for dense matrices whose entries come from a domain R.
Describe a canonical representation for sparse matrices whose entries come from a domain R.
What properties do you require R to have for the previous two parts?
2.
Describe a canonical representation for polynomials in several variables with integer coefficients.
Explain carefully how the representation of previous answer can be used to build a canonical representation for fractions of polynomials in several variables with integer coefficients.
Then give a canonical representation for m*n matrices whose elements are fractions of polynomials in several variables.