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From the Algorithm Fang Cheng to the Matrix Theory M WDS'10 Proceedings of Contributed Papers, Part I, 127–132, 2010. ISBN 978-80-7378-139-2 © MATFYZPRESS From the Algorithm Fang Cheng to the Matrix Theory M. Stˇ ˇep´anov´a Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The aim of the contribution is to trace the origin and development of the matrix theory, which took a long time to become a fully accepted theory and which is closely connected to the study of systems of linear equations, to the theory of determinants and to the theory of bilinear and quadratic forms. We shall list the most important dates in which basic notions of the matrix theory were introduced and of course we shall mention famous mathematicians and their works on this part of linear algebra. Systems of linear equations It is not surprising that the beginning of the matrix theory is connected with the systems of linear equations. Problems which led to easy systems of linear equations were solved in ancient Egypt and Mesopotamia four thousand years ago. A big progress in solution of the system of linear equations occurred two thousand years ago in China. Algorithm fang cheng Chinese mathematicians used an efficient method to solve systems of linear equations. It was called fang cheng and it was written during Han Dynasty in the text Nine Chapters on the Mathematical Art (Jiu zhang suan shu). We will explain the principle of the fang cheng on the next example: There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type? (see [Hudeˇcek, 2008], page 187) This problem led to a system of three linear equations. Chinese mathematicians recorded coefficients of the first equation to the third column, coefficients of the second equation to the second column and coefficients of the third equation to the first column of a rectangular scheme. The Chinese used numerical rods which they put on a counting board to note amounts. We write coefficients of each equation to rows today and then we usually continue to solve a problem by the Gaussian elimination method. The algorithm fang cheng is very similar to this method. The Chinese multiplied the second column by 3 and then subtracted the third column as many times as possible (two times in this case) to get 0 instead of 2 in the second column. Afterwards they did the same with the first column to get 0 instead of 1. The beginning scheme and the scheme after operations referred to above are here: 1 2 3 0 0 3 2 3 2 4 5 2 , . 3 1 1 8 1 1 26 34 39 39 24 39 Subsequently, they tried to get 0 instead of 4 in the left column and so they multiplied theleft column by 5 and subtracted the middle column four times. This process gave the final scheme 127 STˇ EˇPA´NOVA:´ THE ORIGIN OF THE MATRIX THEORY 0 0 3 0 5 2 36 1 1 99 24 39 from which Chinese mathematicians counted the solution for the third type of corn. Next, they used a back substitution to find the solution for the second type of corn and at the end for the first type as well. For other information about the algorithm fang cheng, see [Beˇcv´aˇr, 2007] or [Hudeˇcek, 2008]. The theory of determinants and the theory of bilinear and quadratic forms The theory of determinants is another branch of mathematics which is very closely con- nected to the origin of the theory of matrices. Japanese mathematician Takakazu Sinsukeˇ Seki K¯owa (1642–1708) developed Chinese methods and set together a term similar to a determinant. We usually consider Gottfried Wilhelm Leibniz (1646–1716) to be a discoverer of determinants. He eliminated unknowns from a system of n+1 linear equations in n unknowns and got a term which is called determinant at the present time. He used a couple of numbers for coefficients for the first time. He marked for example by the symbol 32 or 32 the member which we denote a32 today. He also gave the condition that the coefficient matrix must have determinant equal to 0 to be solvable. He wrote this condition for the system of linear equation 10 + 11x + 12y = 0, 20 + 21x + 22y = 0, 30 + 31x + 32y = 0 in this way: 10 · 21 · 32 10 · 22 · 31 11 · 22 · 30 = 11 · 20 · 32 . 12 · 20 · 31 12 · 21 · 30 Origin of the theory of determinants The development of the theory of determinants started after the publication of Cramer’s rule. Scottish mathematician Colin Maclaurin (1698–1746), in the paper Treatise of Algebra in 1748, explained this rule but he did not express it exactly. For example, he did not describe the solution of a system of linear equations with a singular matrix. It was Swiss mathematician Gabriel Cramer (1704–1752) in 1750 who published the monograph about the theory of algebraic curves Introduction `al’analyse des lignes courbes alg´ebriques. There was an accurate explanation of Cramer’s rule in this writing and we consider this work to be the origin of the theory of determinants. More about discussion on the founder of the theory of determinants in [Beˇcv´aˇr, 2007]. The theory of determinants French mathematician Alexandre Th´eophile Vandermonde (1735–1796) published M´emoire sur l’´elimination in 1772. He regarded a determinant as an independent object. His denotation is surprising, he wrote coefficients of a system of linear equations as couples of letters written one above the other without any other symbols. The upper coefficient represented the row and the lower coefficient represented the column. A. T. Vandermonde often used Greek alphabet α for the upper coefficient and Latin alphabet for the lower coefficient. Hence, symbol a denoted what we write as raα. He illustrated determinants with the scheme | | | | 128 STˇ EˇPA´NOVA:´ THE ORIGIN OF THE MATRIX THEORY and introduced some now well-known relationships in this denotation. We state only two of them. The first one describes the finding of the determinant by expanding along the first row: α|β|γ|ϑ α β|γ|ϑ α β|γ|ϑ α β|γ|ϑ α β|γ|ϑ = · − · + · − · . a|b|c|d a b|c|d b c|d|a c d|a|b d a|b|c The second one tells us that the determinant with two identical rows is equal to 0: α|α = 0. a|b German mathematician and astronomer Carl Friedrich Gauss (1777–1855) used in his fa- mous work Disquisitiones arithmeticae in 1801 the square scheme α, β, γ α′, β′, γ′ α′′, β′′, γ′′ to express the linear substitution ′ ′′ x = αy + βy + γy , ′ ′ ′ ′ ′ ′′ x = α y + β y + γ y , ′′ ′′ ′′ ′ ′′ ′′ x = α y + β y + γ y . C. F. Gauss furthermore wrote that the result of the composition of two substitutions with schemes α, β, γ δ, ε, ζ α′, β′, γ′ and δ′, ε′, ζ′ α′′, β′′, γ′′ δ′′, ε′′, ζ′′ is the substitution with the scheme αδ + βδ′ + γδ′′, αε + βε′ + γε′′, αζ + βζ′ + γζ′′ α′δ + β′δ′ + γ′δ′′, α′ε + β′ε′ + γ′ε′′, α′ζ + β′ζ′ + γ′ζ′′ . α′′δ + β′′δ′ + γ′′δ′′, α′′ε + β′′ε′ + γ′′ε′′, α′′ζ + β′′ζ′ + γ′′ζ′′ We can see that this composition represents the matrix multiplication in today sense. C. F. Gauss also denoted the ternary quadratic form ′ ′′ ′ ′ ′ ′′ ′′ ′′ ′ ′′ ′ ′′ ′′ ′ f(x, x , x ) = axx + a x x + a x x + 2bx x + 2b xx + 2b xx briefly by the scheme a a′ a′′ b b′ b′′ and assigned to it the term ′ ′ ′ ′′ ′′ ′′ ′ ′′ ′ ′′ D = abb + a b b + a b b − aa a − 2bb b which he called determinant. We call it discriminant at the present time; it is an opposite value of determinant of the quadratic form f. Gauss’ name is associated with Gaussian elimination which he developed around 1800. Ferdinand Gotthold Max Eisenstein (1823–1852) wrote a matrix of the form ′ ′′ ′ ′ ′ ′′ ′′ ′′ ′ ′′ ′ ′′ ′′ ′ f(x, x , x ) = axx + a x x + a x x + 2bx x + 2b xx + 2b xx 129 STˇ EˇPA´NOVA:´ THE ORIGIN OF THE MATRIX THEORY in his Neue Theoreme der h¨oheren Arithmetik in 1847. For example the form referred to above is associated with the square scheme a b′′ b′ b′′ a′ b . ′ ′′ b b a French mathematician Augustin-Louis Cauchy (1789–1857) denoted determinant of order n in today sense n by the symbol S(±a11a22a33 . ann) and used the term determinant for it for the first time in 1812. Czech mathematician Frantiˇsek Josef Studniˇcka (1836–1903) voiced his opinion that Cauchy was the formal founder of the theory of determinants. At the end, we only list another mathematicians who contributed to the theory of de- terminants: Joseph Louis Lagrange (1736–1813), Pierre Simon Laplace (1749–1827), Jacques Philippe Marie Binet (1786–1856), Carl Gustav Jacob Jacobi (1804–1851), Karl Theodor Weier- strass (1815–1897), Leopold Kronecker (1823–1891) and many others. The origin of the matrix theory Arthur Cayley (1821–1895) and James Joseph Sylvester (1814–1897) played a very impor- tant role in the origin of the matrix theory. They were friends and they worked as lawyers at a court in London. They debated many mathematical problems during their working hours.
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