Page 85 - Matrix Analysis & Applied Linear Algebra

CHAPTER 3

Matrix

Algebra

3.1 FROM ANCIENT CHINA TO ARTHUR CAYLEY

The ancient Chinese appreciated the advantages of array manipulation in dealing
with systems of linear equations, and they possessed the seed that might have
germinated into a genuine theory of matrices. Unfortunately, in the year 213
B.C., emperor Shih Hoang-ti ordered that “all books be burned and all scholars
be buried.” It is presumed that the emperor wanted all knowledge and written
records to begin with him and his regime. The edict was carried out, and it will
never be known how much knowledge was lost. The book Chiu-chang Suan-shu
(Nine Chapters on Arithmetic), mentioned in the introduction to Chapter 1, was
compiled on the basis of remnants that survived.
More than a millennium passed before further progress was documented.
The Chinese counting board with its colored rods and its applications involving
array manipulation to solve linear systems eventually found its way to Japan.
Seki Kowa (1642–1708), whom many Japanese consider to be one of the greatest
mathematicians that their country has produced, carried forward the Chinese
principles involving “rule of thumb” elimination methods on arrays of numbers.
His understanding of the elementary operations used in the Chinese elimination
process led him to formulate the concept of what we now call the determinant.
While formulating his ideas concerning the solution of linear systems, Seki Kowa
anticipated the fundamental concepts of array operations that today form the
basis for matrix algebra. However, there is no evidence that he developed his
array operations to actually construct an algebra for matrices.
From the middle 1600s to the middle 1800s, while Europe was flowering
in mathematical development, the study of array manipulation was exclusively