Linear algebra over finite fields and generic domains in Maple.

Simon Lo, Simon Fraser University

Most linear algebra algorithms are general algorithms that work for matrices over any field. Therefore, it is useful to implement the algorithms in maple such that they are independent of any particular fields and field representations. For some algorithms, it is possible to generalize them further to work for matrices over any Euclidean domain or any integral domain.

In this poster we will show generic Maple code for linear algebra algorithms for matrices over any field, any Euclidean domain and any integral domain. We represent fields and domains in Maple as a table of procedures for arithmetic and other operations. It is also possible to construct certain new domains from existing domains or finite fields. Examples will show computations of Eigenvalues and Eigenvectors over finite fields and computing the Hermite/Smith normal form using the general algorithm over various Euclidean domains. Other examples include Gaussian Elimination, computing Gauss-Jordan form, Hessenberg form, LU decomposition over finite fields, computing the determinant and characteristic polynomial over any integral domain, and other generic linear algebra algorithms.