An Arithmetic for Rectangular Matrix Pencils

This presentation is a generalization of 8] from square, regular n-by-n, pencils to singular and rectangular m-by-n pencils. We deene 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 simpliied description of solutions to discrete-time and continuous-time descriptor systems. A monodromy relation gives a convenient uni-ed characterization of solutions to unforced, discrete descriptor systems that covers both the regular and singular case. An exponential relation (nearly) does the same for continuous-time descriptor systems as well.