Matrix Factorizations

Abstract

The direct solution (by Cramer’s method) of a linear system Mx = b, where $M \in GL_n \left( k \right)\left( {b \in k^n } \right)$ is computationally expensive, especially if one wishes to solve the system many times with various values of b. In the next chapter we shall study iterative methods for the case k=ℝ or ℂ. Here we concentrate on a simple idea: To decompose M as a product PQ in such a way that the resolution of the intermediate systems Py = b and Qx = y is “cheap”. In general, at least one of the matrices is triangular. For example, if P is lower triangular (pij = 0 if i < j), then its diagonal entries pii are nonzero, and one may solve the system Py = b step by step:

$$ \begin{gathered} y_1 = \frac{{b_1 }} {{p_{11} }} \hfill \\ {\text{ }} \vdots \hfill \\ y_i = \frac{{b_i - p_{i1} y_1 - \cdots - p_{i,i} - _1 y_i - _1 }} {{p_{ii} }} \hfill \\ {\text{ }} \vdots \hfill \\ y_n = \frac{{b_n - p_{n1} y_1 - \cdots - p_{n,n - 1} y_{n - 1} }} {{p_{nn} }} \hfill \\ \end{gathered} $$