An Arithmetic for Matrix Pencils

We deene an algebra on matrix pencils that is a natural extension of sums, products and quotients of real numbers. The classical algebra of linear transformations may be regarded as a special case of the algebra of pencils. The sum and product deened here preserve right deeating subspaces. We show below that the matrix sign function and the inverse-free algorithms can be derived from an algebra of linear relations. The linear algebra of relations suggests generalizations and variations of these algorithms.