Modeling with matrices

Abstract

The humble linear map, x↦Ax, and its associated linear system, Ax = b, are so common, so well studied, and so well computed, that their importance can easily slip into the background of computational science. Indeed, most students have likely questioned why they couldn’t simply rely on x = A∖ b as they have studied the numerous ways to solve Ax = b in Chap. 2. The answer is, of course, to gain an appreciation of the intrinsic computational difficulty of solving Ax = b, say by either factoring A or with an iterative method. That said, we note that the coverage in this text on solving Ax = b is but a tiny fraction of the expertise couched in the MATLAB command x = A∖ b. The robust success of this one command speaks for lifetimes of adroit researchers. However, now that this computational power rests with anyone willing to purchase MATLAB or download Octave, one might ask if we can’t simply ignore some of the details and let the mantra that “matrices are important” abate as we gain trust in advanced algorithms and software?.