A Brief History of Linear Algebra and Matrix Theory
The introduction and development of the notion of a matrix and the subject oflinear algebra followed the development of determinants, which arose from the study ofcoefficients of systems of linear equations. Leibnitz, one of the two founders of calculus,
used determinants in 1693 and Cramer presented his determinant-based formula forsolving systems of linear equations (today known as Cramer's Rule) in 1750. In contrast,
the first implicit use of matrices occurred in Lagrange's work on bilinear forms in the late
1700s. Lagrange desired to characterize the maxima and minima of multivariate functions.
His method is now known as the method of Lagrange multipliers. In order to do this hefirst required the first order partial derivatives to be 0 and additionally required that acondition on the matrix of second order partial derivatives hold; this condition is today
called positive or negative definiteness, although Lagrange didn't use matrices explicitly.
Gauss developed Gaussian elimination around 1800 and used it to solve leastsquares problems in celestial computations and later in computations to measure the earth
and its surface (the branch of applied mathematics concerned with measuring or
determining the shape of the earth or with locating exactly points on the earth's surface iscalled geodesy. Even though Gauss' name is associated with this technique forsuccessively eliminating variables from systems of linear equations Chinese manuscripts
from several centuries earlier have been found that explain how to solve a system of threeequations in three unknowns by ''Gaussian'' elimination. For years Gaussian elimination
was considered part of the development of geodesy, not mathematics. The first
appearance of Gauss-Jordan elimination in print was in a handbook on geodesy written
by Wilhelm Jordan. Many people incorrectly assume that the famous mathematician
Camille Jordan is the Jordan in ''Gauss-Jordan'' elimination.
For matrix algebra to fruitfully develop one needed both proper notation and theproper definition of matrix multiplication. Both needs were met at about the same timeand in the same place. In 1848 in England, J.J. Sylvester first introduced the term''matrix,'' which was the Latin word for womb, as a name for an array of numbers. Matrix
algebra was nurtured by the work of Arthur Cayley in 1855. Cayley studied compositionsof linear transformations and was led to define matrix multiplication so that the matrix ofcoefficients for the composite transformation ST is the product of the matrix for S times
the matrix for T. He went on to study the algebra of these compositions including matrix
inverses. The famous Cayley-Hamilton theorem which asserts that a square matrix is aroot of its characteristic polynomial was given by Cayley in his 1858 Memoir on theTheory of Matrices. The use of a single letter A to represent a matrix was crucial to thedevelopment of matrix algebra. Early in the development the formula det(AB) =det(A)det(B) provided a connection between matrix algebra and determinants. Cayley
wrote ''There would be many things to say about this theory of matrices which should, it
seems to me, precede the theory of determinants.''
Mathematicians also attempted to develop of algebra of vectors but there was nonatural definition of the product of two vectors that held in arbitrary dimensions. The firstvector algebra that involved a noncommutative vector product (that is, v x w need notequal w x v) was proposed by Hermann Grassmann in his book Ausdehnungslehre (1844).
Grassmann's text also introduced the product of a column matrix and a row matrix, which
resulted in what is now called a simple or a rank-one matrix. In the late 19th century theAmerican mathematical physicist Willard Gibbs published his famous treatise on vector
analysis. In that treatise Gibbs represented general matrices, which he called dyadics, assums of simple matrices, which Gibbs called dyads. Later the physicist P. A. M. Dirac
introduced the term ''bra-ket'' for what we now call the scalar product of a ''bra'' (row)vector times a ''ket'' (column) vector and the term ''ket-bra'' for the product of a ket times
a bra, resulting in what we now call a simple matrix, as above. Our convention ofidentifying column matrices and vectors was introduced by physicists in the 20th century.
Matrices continued to be closely associated with linear transformations. By 1900
they were just a finite-dimensional subcase of the emerging theory of lineartransformations. The modern definition of a vector space was introduced by Peano in1888. Abstract vector spaces whose elements were functions soon followed.
There was renewed interest in matrices, particularly on the numerical analysis ofmatrices, after World War II with the development of modern digital computers. Johnvon Neumann and Herman Goldstine introduced condition numbers in analyzing roundoff errors in 1947. Alan Turing and von Neumann were the 20th century giants in thedevelopment of stored-program computers. Turing introduced the LU decomposition of amatrix in 1948. The L is a lower triangular matrix with 1's on the diagonal and the U is anechelon matrix. It is common to use LU decompositions in the solution of a sequence ofsystems of linear equations, each having the same coefficient matrix. The benefits of theQR decomposition was realized a decade later. The Q is a matrix whose columns areorthonormal vectors and R is a square upper triangular invertible matrix with positive
entries on its diagonal. The QR factorization is used in computer algorithms for variouscomputations, such as solving equations and find eigenvalues.
References

S. Athloen and R. McLaughlin, Gauss-Jordan reduction: A brief history, American
Mathematical Monthly 94 (1987) 130-142.
A. Tucker, The growing importance of linear algebra in undergraduate mathematics, The
College Mathematics Journal, 24 (1993) 3-9.