The Cayley Hamilton and Frobenius Theorems via the Laplace Transform

The Cayley Hamilton theorem on the characteristic polynomial of a matrix A and Frobenius’ theorem on minimal polynomial of A are deduced from the familiar Laplace transform formula L(eAt) = (sI − A)−1. This formula is extended to a formal power series ring over an algebraically closed field of characteristic 0, so that the argument applies in the more general setting of matrices over a field of characteristic 0. A traditional use of the structure theory of a linear operator has been the calculation of the matrix exponential e and its use in solving the system of differential equations y′ = Ay, when A is a real or complex n× n matrix [2]. Zieber [6] and Schmidt [5] reversed this process and applied knowledge of the basic form of e to a derivation of the main results on the structure of A as a linear operator. Their approach started with an application of the Cayley Hamilton theorem to deduce the form of each entry of e as a solution of a constant coefficient linear differential equation. Since the Cayley Hamilton theorem can be viewed as part of the structure theory of a linear operator, it seems natural to ask if, by means of a different starting point for the analysis of e, one can also deduce this result from information about e. It turns out that the Laplace transform formula [4], (1) L(eAt) = (sI − A)−1 provides such a starting point. This formula applies to real and complex matrices A, but if k is any algebraically closed field of characteristic 0, one can define a Laplace transform on a k-linear subspace of the ring of formal power series k[[t]] in a natural manner so that equation (1) remains valid. We will present our proof of the Cayley Hamilton theorem in this context. Moreover, this approach extends to a simple proof of the Frobenius characterization of the minimal polynomial of 2000 Mathematics Subject Classification. 15A24, 15A54.