R. Checa-Garcia webpage | Linear Algebra

Notebook

Linear Algebra and Computing

Scientific Computing and Linear Algebra

COMPUTING-BLOG
math programming

Table of Contents

Introduction

This post just remember several linear algebra properties that are relevant for scientific programming.

Reducible Matrix

Definition: An square matrix A is reducible if there is a matrix P (basis change) such that, \[PAP^{T}=\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}\]

This property is very useful because the typical linear system solution \(Ay=c\) is now more easy to resolve in terms of computing because \[\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix}=\begin{bmatrix} c_{1}\\ c_{2} \end{bmatrix}\]

The subsystem \(B_{22}y_{2}=c_{2}\) is indepentent of the other B matrices with the logical computational improvement.