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Home , Carl Gustav Jacob Jacobi, Charles Hermite, Eduard Weyr

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 Theory M. Stˇ ˇep´anov´a , Faculty of and Physics, , .

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 , to the theory of 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 and their works on this part of linear .

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 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 Takakazu Sinsukeˇ Seki K¯owa (1642–1708) developed Chinese methods and set together a term similar to a . 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 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 (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

′ ′′ ′ ′ ′ ′′ ′′ ′′ ′ ′′ ′ ′′ ′′ ′ 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 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), (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. In 1841, Arthur Cayley denoted determinant by two vertical lines situated on either side of the square scheme which we use up to the present time to this day. It is interesting that he used many different notations for determinants and matrices. For example he used four vertical lines, two of them on either side of the array, to mark determinants. We usually use this way to notate matrix. Another intriguing fact is that he multiplied matrices by various ways. He sometimes multiplied a column of the first matrix by a column of the second matrix, sometimes a line of the first matrix by a line of the second matrix and sometimes a line of one matrix by a column of the other matrix. It was not surprising at that time because the theory of determinants predominated the theory of matrices and because the determinant of a square matrix is the same as that of its transpose then the multiplication theorem det (A · BT ) = det A · det BT = det A · det B = det (A · B) for determinants was satisfied in all cases. In 1858, A. Cayley published his article A memoir on the theory of matrices in the journal Philosophical Transactions of the Royal Society of London. He motivated an introduction of matrices by an abbreviated notation for a system of linear equations.

Figure 1. An abbreviated notation for a system of linear equations which was used by Arthur Cayley in his article A memoir on the theory of matrices. ([Cayley, 1858], page 17)

130 STˇ EˇPA´NOVA:´ THE ORIGIN OF THE MATRIX THEORY

Arthur Cayley focused on square matrices (especially 2 × 2 and 3 × 3 matrices), he wrote about rectangular matrices only at the end of his work and he discriminated between broad and deep rectangular matrices. . . . the term matrix used without qualification is to be understood as meaning a square matrix. ([Cayley, 1858], page 17) Author regarded matrices as independent objects, he defined matrix addition, multipli- cation, scalar multiplication and other operations, he treated about matrix zero, matrix unity and multiplication by them, he introduced inverse, symmetrical, skew symmetrical, transposed and reciprocal matrix, he defined convertible (LM = ML) and skew convertible (LM = −ML) matrices. Next, Cayley described a relationship between quaternions and matrices. His formulation of the so-called Cayley-Hamilton theorem (he called it general theorem or remarkable theorem), that a matrix satisfies its own characteristic equation, is the most important result of the paper. He proved it for 2 × 2 matrices and indicated some calculations for the case of 3 × 3 matrices but then he wrote . . . but I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree. ([Cayley, 1858], page 24) We usually consider the article A memoir on the theory of matrices to be an origin of the matrix theory. More about discussion on the founder of this part of linear algebra in [Beˇcv´aˇr, 2007]. James Joseph Sylvester, who was very creditable authority at that time, supported Cayley very much. It was Sylvester in 1850 who established the conception of (rectangular) matrix and used the term matrix for the first time in the article Additions to the articles ”On a new class of theorems”, and ”On Pascal’s theorem”. He treated subdeterminants and he defined a nullity of a matrix in 1882. The term nullity of a matrix is closely connected to the term a rank of a matrix. The sum of the rank and the nullity of the square matrix is equal to the order of the matrix. English mathematician Charles Lutwidge Dodgson (1832–1898) is better known under the pseudonym Lewis Carroll as the author of Alice’s adventures in wonderland. He defined a block of a matrix and a minor of a block.

A rank of a matrix English mathematician William Kingdom Clifford (1845–1879) studied zero and non-zero subdeterminants of matrices but he did not define a rank of a matrix. German mathemati- cian Georg Ferdinand Frobenius (1849–1917) performed this notion in two articles in 1879. He defined separately a rank of a square matrix in the first article Uber¨ homogene totale Differen- tialgleichungen: Wenn in einer Determinante alle Unterdeterminanten (m+1)ten Grades verschwinden, die mten Grades aber nicht s¨ammtlich Null sind, so nenne ich m den Rang der Determinante. ([Frobenius, 1879a], Abhandlungen I., page 435) And then he defined a rank of a rectangular matrix in the second article Theorie der linearen Formen mit ganzen Coefficienten:

Gegeben sei ein endliches System A von Gr¨ossen aαβ(α = 1, . . . m; β = 1, . . . n), die nach Zeilen und Colonnen geordnet sind. Wenn in demselben alle Determinanten (l+1)ten Grades verschwinden, die lten Grades aber nicht s¨ammtlich Null sind, so heisst l den Rang des Systems. ([Frobenius, 1879b], Abhandlungen I., page 484)

Canonical forms It is very striking that the study of characteristic and minimal polynomials, real eigenvalues and eigenvectors, the development of canonical forms and orthogonal transformations predated the origin of the matrix theory.

131 STˇ EˇPA´NOVA:´ THE ORIGIN OF THE MATRIX THEORY

The development of canonical form is closely connected with geometry and celestial mechan- ics (characteristic equation was called secular equation in this branch). French mathematician, scientist and philosopher Ren´eDescartes (1596–1650) already studied transformations of equa- tions of conics and quadrics to the sums of squares. Mathematicians , Joseph Louis Lagrange, Carl Gustav Jacob Jacobi and some others tried to transform symmetrical bilinear and quadratic forms to canonical forms in the 18th century. Augustin-Louis Cauchy and French mathematician (1822–1901) proved that eigenvalues of real sym- metric or Hermitian matrices are real numbers. It interrelated to stability of the solar system. James Joseph Sylvester, in A demonstration of the theorem that every homogenous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and neg- ative squares (1852), published Sylvester’s law of inertia. Karl Theodor Wilhelm Weierstrass, Leopold Kronecker and Georg Frobenius formulated their own results concerning the question about canonical forms. Jordan canonical form is named after French mathematician Camille Marie Ennemond Jordan (1838–1922). Czech mathematician (1852–1903) pub- lished his original worldwide reputable results about canonical forms in the eighties of the 19th century.

Conclusion Traces of the matrix theory go back before Christ. The beginnings of this theory are associated with solutions of systems of linear equations and in history are closely connected historically with the theory of determinants and the theory of bilinear and quadratic forms. Both of them are dated before the origin of the theory of matrices in 1858. Moreover, many mathematicians gave priority to determinants in the second half of 19th century and it took a long time than the matrix theory became an independent theory. The study of canonical forms of matrices also predated a research of the matrix theory. For other information about the history of the theory of matrices, see [Beˇcv´aˇr, 2007], [Alten et al., 2003] or [Kline, 1972].

Acknowledgments. The author thanks doc. RNDr. Jindˇrich Beˇcv´aˇr, CSc., for his professional help and incitement to the study. This article was supported by GA CRˇ P401/10/0690 Prameny evropsk´ematematiky.

References Alten, H.-W., A. Djafari Naini, M. Folkerts, H. Schlosser, K.-H. Schlote, H. Wußing, 4000 Jahre Algebra. Geschichte, Kulturen, Menschen, Springer-Verlag, New York, Hongkong, London, Mailand, , Tokio, 2003. Beˇcv´aˇr, J., Z historie line´arn´ıalgebry, Matfyzpress, Praha, 2007. Cayley, A., A Memoir on the Theory of Matrices, Philosophical Transactions of the Royal Society of London, published by The Royal Society, 148 (1858), 17–37. Frobenius, G., Ueber homogene totale Differentialgleichungen, Journal f¨ur die reine und angewandte Mathematik 86, 1–19, 1879a, Abhandlungen I., 435–453. Frobenius, G., Theorie der linearen Formen mit ganzen Coefficienten, Journal f¨ur die reine und ange- wandte Mathematik 86, 146–208, 1879b, Abhandlungen I., 482–544. Hudeˇcek, J., Matematika v dev´ıti kapitol´ach, Matfyzpress, Praha, 2008. Kline, M., Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. Nov´y, L., Origins of Modern Algebra, Academia, Prague, 1973. History topic: Matrices and determinants [online]. Reports available from: http://www-groups.dcs.st- and.ac.uk/∼history/PrintHT/Matrices and determinants.html [2010-05-05].

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