By Olver P.J.
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In this section, we will study inverses of general square matrices. We begin with the formal definition. 1. Let A be a square matrix of size n × n. 7) where I = I n is the n × n identity matrix. The inverse is commonly denoted by X = A −1 . 7) in order to properly define an inverse to the matrix A. The first condition, X A = I , says that X is a left inverse, while the second, A X = I , requires that X also be a right inverse. Rectangular matrices might have either a left inverse or a right inverse, but, as we shall see, only square matrices have both, and so only square matrices 3/15/06 39 c 2006 Peter J.
Before exploring the relevant issues, it will help to reformulate our method in a more convenient matrix notation. 2. Gaussian Elimination — Regular Case. With the basic matrix arithmetic operations in hand, let us now return to our primary task. The goal is to develop a systematic method for solving linear systems of equations. While we could continue to work directly with the equations, matrices provide a convenient alternative that begins by merely shortening the amount of writing, but ultimately leads to profound insight into the structure of linear systems and their solutions.
35) which certainly guarantees that it holds at the solution u . 36) and the resulting iteration scheme is known as Newton’s Method , which, as the name suggests, dates back to the founder of the calculus. To this day, Newton’s Method remains the most important general purpose algorithm for solving equations. It starts with an initial guess u(0) to be supplied by the user, and then successively computes u(k+1) = u(k) − f (u(k) ) . 37) As long as the initial guess is sufficiently close, the iterates u(k) are guaranteed to converge, quadratically fast, to the (simple) root u of the equation f (u) = 0.
AIMS lecture notes on numerical analysis by Olver P.J.