Page 86 - Matrix Analysis & Applied Linear Algebra

80 Chapter 3 Matrix Algebra

dedicated to the theory of determinants. Curiously, matrix algebra did not evolve
along with the study of determinants.
It was not until the work of the British mathematician Arthur Cayley (1821–
1895) that the matrix was singled out as a separate entity, distinct from the
notion of a determinant, and algebraic operations between matrices were defined.
In an 1855 paper, Cayley first introduced his basic ideas that were presented
mainly to simplify notation. Finally, in 1857, Cayley expanded on his original
ideas and wrote A Memoir on the Theory of Matrices. This laid the foundations
for the modern theory and is generally credited for being the birth of the subjects
of matrix analysis and linear algebra.
Arthur Cayley began his career by studying literature at Trinity College,
Cambridge (1838–1842), but developed a side interest in mathematics, which he
studied in his spare time. This “hobby” resulted in his first mathematical paper
in 1841 when he was only 20 years old. To make a living, he entered the legal
profession and practiced law for 14 years. However, his main interest was still
mathematics. During the legal years alone, Cayley published almost 300 papers
in mathematics.
In 1850 Cayley crossed paths with James J. Sylvester, and between the two
of them matrix theory was born and nurtured. The two have been referred to
as the “invariant twins.” Although Cayley and Sylvester shared many mathe-
matical interests, they were quite different people, especially in their approach
to mathematics. Cayley had an insatiable hunger for the subject, and he read
everything that he could lay his hands on. Sylvester, on the other hand, could
not stand the sight of papers written by others. Cayley never forgot anything
he had read or seen—he became a living encyclopedia. Sylvester, so it is said,
would frequently fail to remember even his own theorems.
In 1863, Cayley was given a chair in mathematics at Cambridge University,
and thereafter his mathematical output was enormous. Only Cauchy and Euler
were as prolific. Cayley often said, “I really love my subject,” and all indica-
tions substantiate that this was indeed the way he felt. He remained a working
mathematician until his death at age 74.
Because the idea of the determinant preceded concepts of matrix algebra by
at least two centuries, Morris Kline says in his book Mathematical Thought from
Ancient to Modern Times that “the subject of matrix theory was well developed
before it was created.” This must have indeed been the case because immediately
after the publication of Cayley’s memoir, the subjects of matrix theory and linear
algebra virtually exploded and quickly evolved into a discipline that now occupies
a central position in applied mathematics.