A Study on the Jordan Canonical Form in vector spaces over finite fields

Abstract

In this work, we will present the construction of the theorem that substantiates the existence of the Jordan Canonical Form for a given linear transformation on a vector space over and algebraically closed field using the fact that, if 𝑇 is a linear transformation in a vector space 𝑉 over a field K and 𝜆 is an eigenvalue of 𝑇, then the restriction of 𝑇 − 𝜆_𝑖 to ker(𝑇 − 𝜆𝐼𝑑)^𝑗 (where 𝑗 is the smallest integer such that ker(𝑇 − 𝜆𝐼𝑑)^𝑗 = ker(𝑇 − 𝜆𝐼𝑑)^𝑗+1) is nilpotent, which allows us to find a set of vectors 𝛽={𝑣, (𝑇 − 𝜆𝐼𝑑)(𝑣), . . . , (𝑇 − 𝜆𝐼𝑑)^𝑚−1(𝑣)} such that the restriction of 𝑇 − 𝜆𝐼 to the set generated by 𝛽 can be represented by a matrix of the type 𝐽_𝑚(0), therefore, the restriction of 𝑇 to [𝛽] can be represented by 𝐽_𝑚(𝜆_𝑖).
Following that, we will use Field Theory to extend such construction in order to allow it to be used in vector spaces over finite fields, since if 𝑝(𝑥) is an irreducible polynomial over a field K, then K[𝑥]\⟨𝑝(𝑥)⟩ is a field where 𝑝(𝑥) has roots.