If one's interest lies in solving differential equations or in calculating matrix exponentials, there appears to be a wealth of material.
Besides the references already described, we could mention a more complete follow-on to the articles of Putzer, Kirchner, and Leonard which was published by Saber N. Elaydi and William A. Harris, Jr. in 1998 [28] ``On the computation of AN.'' It explicitly mentions the resolution of the identity via partial fraction decomposition of equation W, and uses the Casorati matrix, which replaces Luz' Wronskian in the context of finite differences.
In 1975 Edward P. Fulmer [29] published ``Computation of the matrix exponential,'' citing Kirchner as well as a still earlier reference, and showing how confluent Vandermonde matrices can be involved in the solution. In the article immediately following, ``Explicit formulas for solutions of the second-order matrix differential equation Y''=AY,'' Tom M. Apostol, [30] adapts the calculation to this frequently occurring equation.
Eugene Garfield's Institute for Scientific Information not only publishes the Citation Index, but offers to conduct searches and make reference trees for keywords selected by their clients. One could probably have an interesting time tracing down such keys as Putzer's method or matrix exponential, but the literature which we have encountered in our own library is probably a good sampling of the kinds of articles and their contents.
The subject is a popular one, and seems to be getting to be more so, judging by Java applets to be found on the Internet, and its inclusion in symbolic algebra systems such as Macsyma, Maple, or Mathematica. Looking in the opposite direction, one wonders what there is to be found in the literature of the first half of the century. The standard reference on matrix theory was Frazer, Duncan and Collar's Elementary Matrices, whose very title indicates an inclusion of general, and not just normal, matrices. However, it is only a fond memory, seeming to be both out of print and not to be found in the libraries to which we have access.
Those authors were aeronautical engineers and concerned with mechanical vibrations. Engineering problems are prone to encounter the Jordan form; one has only to recall that the case of critical damping is the important one for the vibration of mechanical systems. That means that one should probably look to the engineering, rather than quantum mechanical, literature for sympathetic discussions of matrices in their full generality (notwithstanding the fact that in parameter space, the ``bad'' matrices have measure zero). In that direction, a series of books by I. Gohberg (and in particular the latest [26], which builds on its predecessors), which concentrates on finite polynomials involving finite matrices, is worth consulting.
Of course, there are more traditional books on matrix theory, of which a classic is surely F. R. Gantmacher's The Theory of Matrices, [31]. Also worthy of note is A. I. Mal'cev's Foundations of Linear Algebra, [27]. They give very careful and exhaustive analyses of the normal forms. Unfortunatley it is just that attention to detail which has alienated the authors of all these other papers which wee have cited, which in turn, by concentrating exclusively on the matrix exponential, have obscured a much more general result.