Leveraging the Cayley Hamilton Theorem for Efficiently Solving the Jordan Canonical Form Problem

TL;DR

This paper introduces an efficient method leveraging the Cayley-Hamilton Theorem to compute the Jordan Canonical Form by avoiding expensive kernel calculations, resulting in faster determination of Jordan Chains.

Contribution

It presents a novel approach to find starting vectors for Jordan Chains without kernel computations, improving efficiency in Jordan Canonical Form calculation.

Findings

Method reduces computational complexity.

Proves maximal length of Jordan Chains.

Provides a new algorithm for Jordan form computation.

Abstract

Given an $n \times n$ nonsingular matrix A and the characteristic polynomial of A as the starting point, we will leverage the Cayley-Hamilton Theorem to efficiently calculate the maximal length Jordan Chains for each distinct eigenvalue of the matrix. Efficiency and speed are gained by seeking a certain type of starting vector as the first step of the algorithm. The method for finding this starting vector does not require calculating the $ker[(A-\lambda I)^k]$ which is quite an expensive operation, and which is the usual approach taken in solving the Jordan Canonical basis problem. Given this starting vector, all remaining vectors in the Jordan Chain are calculated very quickly in a loop. The vectors comprising the Jordan Chains will then be used to minimize the amount of equation solving in order to find the remaining generalized eigenvector basis. We will prove a theorem that justifies why the resulting Jordan Chains are of maximal length. We will also justify how to derive the starting vector which will subsequently be used to calculate the Maximal Jordan Chains.