VECTORS and MATRICES by Cyrus Colton McDuffie

Sixty years ago C.C. McD. wrote a very small book on matrix theory, which was less than a century old at the time. His Introduction contained a sketch of the early historical development of the subject, which I think gives an interesting perspective on the whole subject. Plus he has a writing style that’s concise and humorous, as far as the history of matrix theory can be funny. Here’s some of it.

INTRODUCTION

The theory of matrices had its origin in the theory of determinants, and the latter had its origins in the theory of systems of equations. From Vandermonde and Laplace to Cayley, determinants were cultivated in a purely formal manner. The early algebraists never successfully explained what a determinant was, and indeed they were not interested in exact definitions.

It was Cayley who seems first to have noticed that “the idea of matrix precedes that of determinant”. More explicitly, we can say that the relation of determinant* to matrix is that of the absolute value of a complex number to the complex number itself, and it is no more possible to define determinant without the previous concept of matrix than it is to have the feline grin without the Cheshire cat.

In fact, the importance of the concept of determinant has been, and currently is, vastly over-estimated. Systems of equations can be solved as easily and neatly without determinants as with, as is illustrated in Chapter I of this monograph. In fact, perhaps ninety per cent of matric theory can be developed without mentioning a determinant. The concept is necessary in some places, however, and is very useful in many others, so one should not push this point too far.

In the middle of the last century matrices were approached from several different points of view. The paper of Hamilton (1853) on “Linear and vector functions” is considered by Wedderburn to contain the beginnings of the theory. After developing some properties of “linear transformations” in earlier papers, Cayley finally wrote “A Memoir on the Theory of Matrices”, in which a matrix is considered as a single mathematical quantity. This paper gives Cayley considerable claim to the honor of introducing the modern concept of matrix, although the name is due to Sylvester (1850).

In 1867 there appeared the beautiful paper of Laguerre entitled “Sur le calcul des systemes lineaires” in which matrices are treated in almost the modern manner. It attracted little attention at the time of its publication. Frobenius, in his fundamental paper “Ueber lineare Substitutionen und bilinearen Formen” of 1878, approached matric theory throught the composition of quadratic forms.

In fact, Hamilton, Cayley, Laguerre, and Frobenius seem to have worked without the knowledge of each others’ results. Frobenius, however, very soon became aware of these earlier papers and eventually adopted the term “matrix”.

*Note: in the first half of the 20^th century the concept of matrix norm was not widely used, but from our present vantage point McDuffie’s remark is really more applicable to norms than determinants.