Application of some explicit formulas for the matrix exponential in linear systems software validation

While we offered the first link between idempotent matrices and expedient easy calculation of the matrix exponential for pre-validation software testing in 1989, there have been several interesting closed-form expressions subsequently offered by others for the matrix exponential exp(F) under special cases of the system matrix F being merely 2×2 or 3×3 or else satisfying fairly restrictive quadratic or cubic equality constraints (inherited somehow from particular applications?) and various closed-form results have also been offered for particular special cases of an n×n F being either nilpotent, involute, or idempotent. Although practical physical systems exhibiting these particular structural properties seldom arise in practice, we alert the reader to a variety of useful applications of these analytic closed-form results and their immediate extensions offered herein to the verification/validation of control and estimation related software. In this same vein, we call attention to other expressions that arise in general systems theory and its simulation that simplify nicely to closed-form for idempotent matrices. We also mention other related precedents and extensions beyond those previously covered regarding skew-symmetric rotation matrices arising in the role of the system matrix and discuss algorithms for computing exp(Ft) from a finite matrix series in the completely general n×n case