This theory provides a compact formulation of Gauss-Jordan elimination for matrices represented as functions. Its distinctive feature is succinctness. It is not meant for large computations. 1 Gauss-Jordan elimination algorithm theory Gauss-Jordan-Elim-Fun imports Main begin Matrices are functions: type-synonym ′a matrix = nat ⇒ nat ⇒ ′a In order to restrict to finite matrices, a matrix is usua...

This paper outlines a proof of the Jordan Normal Form Theorem. First we show that a complex, finite dimensional vector space can be decomposed into a direct sum of invariant subspaces. Then, using induction, we show the Jordan Normal Form is represented by several cyclic, nilpotent matrices each plus an eigenvalue times the identity matrix – these are the Jordan

The theorems of this paper show that the main results in the structure and representation theory of Jordan algebras and of alternative algebras are valid for a larger class of algebras defined by simple identities which obviously hold in the Jordan and alternative cass. A new unification of the Jordan and associative theories is also achieved.