The central reference in this whole discussion was published by E. J. Putzer [18] in the American Mathematical Monthly in 1966 while he was working for an aircraft manufacturer. The title of the article, ``Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients,'' states its intention quite concisely and in retrospect, the paper may have been composed more from an engineering background than one in physics and mathematics. The reason for this suspicion is that engineers are much more familiar with nonnormal matrices, which are never encountered when physicists use hermitean operators.
His paper is confined to the matrix exponential, making its point by showing that the solution of a differential equation of a certain order actually generates the required exponential. It is extremely easy to read too much into this procedure, which was the source of several months' confusion. Namely, the differential equation which is solved is the one worthy of the exponential of the companion matrix, not the matrix itself. The missed point: the first result is not in itself the desired exponential, but part of a two stage process which first generates the coefficients for the power series representation of the actual representation. The second step can include a matrix inversion and invites some subsidiary discussion; but it is important to realize that there are two steps, and that the Jordan normal form in the companion matrix majorizes the Jordan normal forms for all other matrices having the same symmetric functions.