Matrix (mathematics) - History

History

Matrices have a long history of application in solving linear equations. The Chinese text The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations, including the concept of determinants, over 1000 years before its publication by the Japanese mathematician Seki in 1683 and the German mathematician Leibniz in 1693. Cramer presented his rule in 1750.

Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices. The term "matrix" (Latin for "womb", derived from mater—mother) was coined by Sylvester in 1850, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:

I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.

The study of determinants sprang from several sources. Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = the following: replace the powers a_jk by a_jk in the polynomial

,

where Π denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real. Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten and Weierstrass' Zur Determinantentheorie, both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.

Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra. partially due to their use in classification of the hypercomplex number systems of the previous century.

The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns. Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.