Arithmetic for rectangular matrix pencils

This paper is a generalization of the authors' (1998) previous study from square, regular n-by-n pencils to singular and rectangular m-by-n pencils. We define arithmetic-like operations on matrix pencils that are a natural extension of sums, products and quotients of real numbers. The algebra of linear transformations may be regarded as a special case of this pencil arithmetic. The language of linear relations leads to an inverse free matrix sign function algorithm and gives a simplified description of solutions to discrete-time and continuous-time descriptor systems. A monodromy relation gives a convenient unified characterization of solutions to unforced, discrete descriptor systems that covers both the regular and singular cases. An exponential relation (nearly) does the same for continuous-time descriptor systems as well.