Super Linear Algebra

TL;DR

This book introduces super linear algebra and super vector spaces, extending traditional linear algebra with super matrices that can store data in multiple blocks, offering new tools for computational applications.

Contribution

It formalizes super linear algebra and super vector spaces, providing foundational theorems, properties, and applications in fields like Markov chains and economic models.

Findings

Super matrices enable multi-block data storage.

Theorems on super linear algebra properties are established.

Applications demonstrated in Markov chains and economic models.

Abstract

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized world. Some other structures ought to replace linear algebras which are over a century old. Super linear algebras that use super matrices can store data not only in a block but in multiple blocks so it is certainly more powerful than the usual matrices. This book has 3 chapters. Chapter one introduces the notion of super vector spaces and enumerates a number of properties. Chapter two defines the notion of super linear algebra, super inner product spaces and super bilinear forms. Several interesting properties are derived. The main application of these new structures in Markov chains and Leontief economic models are also given in this chapter. The final chapter suggests 161 problems mainly to make the reader understand this new concept and apply them.