An Arithmetic for Matrix Pencils
Peter Benner, Ralph E. Byers · 1998
We define 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 defined here preserve right deflating 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. 1 Linear Relations A linear relation is a set of ordered pairs of the form RE;A = f(x; y) 2 R n \\Theta R n j Ey = Axg (1) for some matrices E 2 R m\\Thetan and A 2 R m\\Thetan and m 2n. The representation of the linear relation (1) in terms of E and A is not unique. If M 2 R m\\Thetam is nonsingular, then RE;A = RME;MA . If E has full column rank n, then the relation is a linear transformation represented by y = (E T E) \\Gamma1 E T Ax. From another point of view...