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1 Introduction
 1.1 What is the role of the MatricesForHomalg package in the homalg project?
 1.2 This manual

1 Introduction

1.1 What is the role of the MatricesForHomalg package in the homalg project?

1.1-1 MatricesForHomalg provides ...

The package MatricesForHomalg provides:

1.1-2 homalg delegates ...

The package homalg delegates all matrix operations as it treats matrices and their rings as black boxes. homalg comes with a single predefined class of rings and a single predefined class of matrices over these rings -- the so-called internal matrices (--> 5.1-2) over so-called internal rings (--> 3.1-4). An internal matrix (resp. ring) is simply a wrapper containing a GAP-builtin matrix (resp. ring). homalg allows other packages to define further classes or extend existing classes of rings and matrices together with their operations. For example:

The matrix operations are divided into two classes called "Tools" and "Basic". The "Tools" operations include addition, subtraction, multiplication, extracting certain rows or columns, stacking, and augmenting matrices (--> Appendix B). The "Basic" operations include the two basic operations in linear algebra needed to solve an inhomogeneous linear system \(XA=B\) with coefficients in a not necessarily commutative ring \(R\) (--> Appendix A):

The first operation is nothing but deciding the solvability of the inhomogeneous system \(XA=B\) and if solvable to compute a particular solution \(X\), while the second is to compute an \(R\)-generating set for the homogeneous solution space, i.e. the solution space of the homogeneous system \(YA=0\). The above is of course also valid for the column convention.

1.1-3 The black box concept

Now we address the following concerns: Wouldn't the idea of using algorithms like the Gröbnerbasis algorithm(s) as a black box (--> homalg delegates ...) contradict the following facts?

The following should clarify the above concerns.

1.2 This manual

Chapter 2 describes the installation of this package. The remaining chapters are each devoted to one of the MatricesForHomalg objects (--> 1.1-1) with its constructors, properties, attributes, and operations.

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