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Given an $n \times n$ nonsingular matrix A and the characteristic polynomialof A as the starting point, we will leverage the Cayley-Hamilton Theorem toefficiently calculate the maximal length Jordan Chains for each distincteigenvalue of the matrix. Efficiency and speed are gained by seeking a certaintype of starting vector as the first step of the algorithm. The method forfinding this starting vector does not require calculating the $ker[(A-\lambdaI)^k]$ which is quite an expensive operation, and which is the usual approachtaken in solving the Jordan Canonical basis problem. Given this startingvector, all remaining vectors in the Jordan Chain are calculated very quicklyin a loop. The vectors comprising the Jordan Chains will then be used tominimize the amount of equation solving in order to find the remaininggeneralized eigenvector basis. We will prove a theorem that justifies why theresulting Jordan Chains are of maximal length. We will also justify how toderive the starting vector which will subsequently be used to calculate theMaximal Jordan Chains.

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