ࡱ> EGBCD@ ObjbjFF ",,OYYYYZl!z>[ eeeef6jylyyyyyyy$_{R}yoffooyeeyNxNxNxoeeyNxoyNxNxNxeZ u%YHuNxyy0!zNx=~Jw=~NxNx&=~ty,Amm|NxkndnAmAmAmyy*.*Dx . Matrix Algebra History: ( HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html" http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html) The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway. It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:- There are two fields whose total area is 1800 square yards. One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Nine_chapters.html" Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:- 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? Now the author does something quite remarkable. He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board'. 1 2 3 2 3 2 3 1 1 26 34 39 Our late 20th Century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical. Most remarkably the author, writing in 200 BC, instructs the reader to multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives 0 0 3 4 5 2 8 1 1 39 24 39 Next the left most column is multiplied by 5 and then the middle column is subtracted as many times as possible. This gives 0 0 3 0 5 2 36 1 1 99 24 39 from which the solution can be found for the third type of corn, then for the second, then the first by back substitution. This method, now known as Gaussian elimination, would not become well known until the early 19th Century.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cardan.html" Cardan, in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [7] calls mother of rules ! This rule gives what essentially is  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer's rule for solving a 2  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 2 system although  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cardan.html" Cardan does not make the final step.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cardan.html" Cardan therefore does not reach the definition of a determinant but, with the advantage of hindsight, we can see that his method does lead to the definition. Many standard results of elementary matrix theory first appeared long before matrices were the object of mathematical investigation. For example  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_Witt.html" de Witt in Elements of curves, published as a part of the commentaries on the 1660 Latin version of  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html" Descartes' Gomtrie , showed how a transformation of the axes reduces a given equation for a conic to canonical form. This amounts to diagonalising a symmetric matrix but  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_Witt.html" de Witt never thought in these terms. The idea of a determinant appeared in Japan and Europe at almost exactly the same time although  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html" Seki in Japan certainly published first. In 1683  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html" Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the way the Chinese methods described above were constructed. Without having any word which corresponds to 'determinant'  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html" Seki still introduced determinants and gave general methods for calculating them based on examples. Using his 'determinants'  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html" Seki was able to find determinants of 2  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 2, 3  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 3, 4  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 4 and 5  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 5 matrices and applied them to solving equations but not systems of linear equations. Rather remarkably the first appearance of a determinant in Europe appeared in exactly the same year 1683. In that year  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz wrote to  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html" de l'Hpital. He explained that the system of equations 10 + 11x + 12y = 0 20 + 21x + 22y = 0 30 + 31x + 32y = 0 had a solution because 10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30 which is exactly the condition that the coefficient matrix has determinant 0. Notice that here  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz is not using numerical coefficients but two characters, the first marking in which equation it occurs, the second marking which letter it belongs to. Hence 21 denotes what we might write as a21.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz was convinced that good mathematical notation was the key to progress so he experimented with different notation for coefficient systems. His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678. Only two publications (1700 and 1710) contain results on coefficient systems and these use the same notation as in his letter to  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html" de l'Hpital mentioned above.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace expansion. As well as studying coefficient systems of equations which led him to determinants,  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory. In the 1730's  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html" Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer's rule for 2  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 2 and 3  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 3 systems and indicating how the 4  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 4 case would work.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer gave the general rule for n  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET n systems in a paper Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. The rule appears in an Appendix to the paper but no proof is given:- One finds the value of each unknown by forming n fractions of which the common denominator has as many terms as there are permutations of n things.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign. He also says how the n numerators of the fractions can be found by replacing certain coefficients in this calculation by constant terms of the system. Work on determinants now began to appear regularly. In 1764  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bezout.html" Bezout gave methods of calculating determinants as did  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html" Vandermonde in 1771. In 1772  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace claimed that the methods introduced by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cramer.html" Cramer and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bezout.html" Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of systems of linear equations without actually calculating it, by using determinants. Rather surprisingly  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace used the word 'resultant' for what we now call the determinant: surprising since it is the same word as used by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz yet  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace must have been unaware of  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html" Leibniz's work.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace gave the expansion of a determinant which is now named after him.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html" Lagrange, in a paper of 1773, studied identities for 3  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 3 functional determinants. However this comment is made with hindsight since  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html" Lagrange himself saw no connection between his work and that of  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html" Laplace and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html" Vandermonde. This 1773 paper on mechanics, however, contains what we now think of as the volume interpretation of a determinant for the first time.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html" Lagrange showed that the tetrahedron formed by O(0,0,0) and the three points M(x,y,z), M'(x',y',z'), M"(x",y",z") has volume 1/6 [z(x'y" - y'x") + z'(yx" - xy") + z"(xy' - yx')]. The term 'determinant' was first introduced by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html" Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic form. However the concept is not the same as that of our determinant. In the same work  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html" Gauss lays out the coefficients of his quadratic forms in rectangular arrays. He describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra) and the inverse of a matrix in the particular context of the arrays of coefficients of quadratic forms. Gaussian elimination, which first appeared in the text  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Nine_chapters.html" Nine Chapters on the Mathematical Art written in 200 BC, was used by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html" Gauss in his work which studied the orbit of the asteroid Pallas. Using observations of Pallas taken between 1803 and 1809,  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html" Gauss obtained a system of six linear equations in six unknowns.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html" Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix. It was  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy in 1812 who used 'determinant' in its modern sense.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy's work is the most complete of the early works on determinants. He reproved the earlier results and gave new results of his own on minors and adjoints. In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France,  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Binet.html" Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy. In 1826  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients. He found the eigenvalues and gave results on diagonalisation of a matrix in the context of converting a form to the sum of squares.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation. He also, again in the context of quadratic forms, proved that every real symmetric matrix is diagonalisable. Jacques  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html" Sturm gave a generalisation of the eigenvalue problem in the context of solving systems of ordinary differential equations. In fact the concept of an eigenvalue appeared 80 years earlier, again in work on systems of linear differential equations, by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html" D'Alembert studying the motion of a string with masses attached to it at various points. It should be stressed that neither  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cauchy.html" Cauchy nor Jacques  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html" Sturm realised the generality of the ideas they were introducing and saw them only in the specific contexts in which they were working.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html" Jacobi from around 1830 and then  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html" Kronecker and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html" Weierstrass in the 1850's and 1860's also looked at matrix results but again in a special context, this time the notion of a linear transformation.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html" Jacobi published three treatises on determinants in 1841. These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions. These three papers by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html" Jacobi made the idea of a determinant widely known.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley, also writing in 1841, published the first English contribution to the theory of determinants. In this paper he used two vertical lines on either side of the array to denote the determinant, a notation which has now become standard.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html" Eisenstein in 1844 denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers except for the lack of commutativity. It is fair to say that  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html" Eisenstein was the first to think of linear substitutions as forming an algebra as can be seen in this quote from his 1844 paper:- An algorithm for calculation can be based on this, it consists of applying the usual rules for the operations of multiplication, division, and exponentiation to symbolic equations between linear systems, correct symbolic equations are always obtained, the sole consideration being that the order of the factors may not be altered. The first to use the term 'matrix' was  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester in 1850.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it. After leaving America and returning to England in 1851,  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester became a lawyer and met  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley, a fellow lawyer who shared his interest in mathematics.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley quickly saw the significance of the matrix concept and by 1853  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley had published a note giving, for the first time, the inverse of a matrix.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley in 1858 published Memoir on the theory of matrices which is remarkable for containing the first abstract definition of a matrix. He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses. He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley also proved that, in the case of 2  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 2 matrices, that a matrix satisfies its own characteristic equation. He stated that he had checked the result for 3  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 3 matrices, indicating its proof, but says:- 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. That a matrix satisfies its own characteristic equation is called the  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley- HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html" Hamilton theorem so its reasonable to ask what it has to do with  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html" Hamilton. In fact he also proved a special case of the theorem, the 4  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 4 case, in the course of his investigations into quaternions. In 1870 the  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Jordan.html" Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan. It appears in the context of a canonical form for linear substitutions over the finite field of order a prime.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius, in 1878, wrote an important work on matrices On linear substitutions and bilinear forms although he seemed unaware of  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley's work.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius in this paper deals with coefficients of forms and does not use the term matrix. However he proved important results on canonical matrices as representatives of equivalence classes of matrices. He cites  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html" Kronecker and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html" Weierstrass as having considered special cases of his results in 1874 and 1868 respectively.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius also proved the general result that a matrix satisfies its characteristic equation. This 1878 paper by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius also contains the definition of the rank of a matrix which he used in his work on canonical forms and the definition of orthogonal matrices. The nullity of a square matrix was defined by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester in 1884. He defined the nullity of A, n(A), to be the largest i such that every minor of A of order n-i+1 is zero.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester was interested in invariants of matrices, that is properties which are not changed by certain transformations.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html" Sylvester proved that max{n(A), n(B)}  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/lte.gif" \* MERGEFORMATINET n(AB)  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/lte.gif" \* MERGEFORMATINET n(A) + n(B). In 1896  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius became aware of  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley's 1858 Memoir on the theory of matrices and after this started to use the term matrix. Despite the fact that  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley only proved the  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley- HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html" Hamilton theorem for 2  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 2 and 3  INCLUDEPICTURE "http://www-groups.dcs.st-and.ac.uk/~history/Symbolgifs/cross.gif" \* MERGEFORMATINET 3 matrices,  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius generously attributed the result to  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html" Cayley despite the fact that  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html" Frobenius had been the first to prove the general theorem. An axiomatic definition of a determinant was used by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html" Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory. In the same year  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html" Kronecker's lectures on determinants were also published, again after his death. With these two publications the modern theory of determinants was in place but matrix theory took slightly longer to become a fully accepted theory. An important early text which brought matrices into their proper place within mathematics was Introduction to higher algebra by  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bocher.html" Bcher in 1907.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html" Turnbull and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aitken.html" Aitken wrote influential texts in the 1930's and  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mirsky.html" Mirsky's An introduction to linear algebra in 1955 saw matrix theory reach its present major role in as one of the most important undergraduate mathematics topic.  HYPERLINK "http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/References/Matrices_and_determinants.html" \t "_blank" References (13 books/articles) Article by: J J O'Connor and E F Robertson An introduction to the theory and application of matrices V1.1 January 2002 ( HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" http://www.morello.co.uk/matrixalgebra.htm) This document is copyright Morello Publishing Ltd 2001-2, all rights reserved worldwide. The latest version of this document is available from  HYPERLINK "http://www.morello.co.uk/" http://www.morello.co.uk free of charge. You are permitted to distribute this document via the Internet, CD-ROM or any other media provided it is complete and unmodified including this copyright notice and URL. 1 Introduction This document serves as an introduction to the basics of matrix algebra, and the application of matrices to solving common problems. We produced this guide primarily to support of MatrixCalculator software for Windows, but we hope it will be useful to others too. You can download a trial version of the program from our website,  HYPERLINK "http://www.morello.co.uk/" www.morello.co.uk What is a matrix? A matrix is a rectangular array of numbers (called elements), consisting of m rows and n columns:  This is said to be a matrix of order m by n. For instance, here is a 2 by 3 matrix:  If you are from a C/C++ programming background, you will notice that a matrix is very much like a 2 dimensional array, but beware the indices start from 1 (whereas C array indices start from zero). Also, note that the ordering is row index followed by column index, where a programmer might naturally put the column index first. This is just the way matrix algebra is traditionally written. Matrices are useful in a solving a number of problems. After we have described the basic algebra of matrices in the sections below, we will go on to give a couple of real life examples. 2 Matrix Algebra Addition It is only possible to add 2 matrices if they are of the same order (ie same number of rows and columns). If we add matrices A and B to give a result X, then each element of X is simply the sum of the 2 corresponding elements of A and B,  for example  Since matrix addition is performed simply by adding the individual elements, clearly you will get the same result whatever order you add the matrices in. In maths jargon we say that the operation is both commutative and associative:  Subtraction Subtraction of 2 matrices is analogous with  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Addition#_Addition" addition:  for example, using the same A and B as before  Scalar Multiplication Scalar multiplication, ie multiplying a matrix by a number (eg F) is simply a matter of multiplying each element of the matrix by the number:  For example:  It should be clear from the above that scalar multiplication is commutative, ie  and also that scalar multiplication is distributive over  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Addition#_Addition" addition and  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Subtraction#_Subtraction" subtraction, ie  Matrix Multiplication We can multiply 2 matrices to give a matrix result:  It is only possible to multiply A and B if the number of columns of A is equal to the number of rows of B (A and B are then said to be conformable). If A is an n by m matrix, and B is an m by p matrix, then X will be a n by p matrix. The definition of X is:  For example, we can multiply a 2 by 3 matrix and a 3 by 2 matrix, resulting in a 2 by 2 matrix:  Notice what happens if we change the order of the 2 matrices. This time we are multiplying a 3 by 2 matrix with a 2 by 3 matrix, and the result is a 3 by 3 matrix, quite different from the previous result:  This illustrates that matrix multiplication is not commutative. In fact, if you exchange the matrices, the multiplication may become invalid due to conformability. For example, if A is a 2 by 3 matrix and B is a 3 by 3 matrix, it is possible to for the product AB, but not BA. Matrix multiplication is, however, associative and distributive. In summary:  Transposition Transposing a matrix means converting and m by n matrix into an n by m matrix, by flipping the rows and columns.  It is denoted by a superscript T, eg:  As an aside, there is an interesting relationship between transposition and  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Matrix_Multiplication#_Matrix_Multiplication" multiplication:  If you are interested, you can prove this for yourself fairly easily. Hint look at the definition of matrix multiply, and try swapping the subscripts! Equality 2 matrices are considered to be equal if they are of the same order, and if all their corresponding elements are equal. 3 Special Types of Matrix Vector A row vector is a matrix containing a single row, eg  A column vector is a matrix containing a single column, eg  Both of these forms can be used to represent vector quantities which can be manipulated by matrix algebra, see later. To convert between these forms, simply  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Transposition#_Transposition" transpose the matrix. Zero (Null) Matrix A zero, or null, matrix is one where every element is zero, eg  Square Matrix A square matrix is one where the number of rows and columns are equal, eg a 2 by 2 matrix, a 3 by 3 matrix etc. Diagonal Matrix A diagonal matrix is a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix in which all the elements are zero except for the elements on the leading diagonal, eg:  Unit Matrix A unit matrix is a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix in which all the elements on the leading diagonal are 1, and all the other elements are 0, eg:  The reason that this is called a unit matrix is that if you multiply any matrix by a unit matrix (of the correct size), you will get back the original matrix. A unit matrix is often denoted by I (identity matrix). Symmetric Matrix A symmetrix matrix is a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix where  for all elements. Ie, the matrix is symmetrical about the leading diagonal. For example  Skew Symmetric Matrix A skew symmetrix matrix is a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix where  for all elements. Ie, the matrix is anti-symmetrical about the leading diagonal. This of course requires that elements along the diagonal must be zero. For example  Orthogonal Matrix An orthogonal matrix is a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix which produces a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Unit_Matrix#_Unit_Matrix" unit matrix if it is multiplied by its own  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Transposition#_Transposition" transpose. Ie:  4 Inverse Matrices and Determinants The Inverse of a Matrix The inverse (or reciprocal) of a square matrix is denoted by the A-1, and is defined by  For example  The 2 matrices on the left are inverses of each other, whose product is the  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Unit_Matrix#_Unit_Matrix" unit matrix. Not all matrices have an inverse, and those which dont are called singular matrices. Inverting a matrix is a very useful technique, and we will see, but how is it done? Unfortunately it is slightly more complicated than the basic matrix algebra of the previous chapters, so we will need to take a slight detour into the areas of determinants, cofactors and adjoint matrices first. Determinants In this section we are simply going to define the determinant and, in later sections, point out some of its properties. A deeper discussion of determinants probably deserves its own paper. The determinant of a  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix is a single number calculated by combining all the elements of the matrix. For example, the determinant of a 2 by 2 matrix is  For a 3 by 3 matrix the formula is  The 2 by 2 determinants are called minors. Every element in a determinant has a corresponding minor, formed by deleting the row and column containing that element. For a determinant of order n, the minors are of order (n-1). In general a determinant of order n is calculated from  where m is the minor of a. Cofactors The cofactor of an element is the minor multiplied by the appropriate sign  or more generally  Adjoint Matrices Every  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Square_Matrix#_Square_Matrix" square matrix has an adjoint matrix, found by taking the matrix of its  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Cofactors#_Cofactors" cofactors, and  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Transposition#_Transposition" transposing it, ie if  then the adjoint is  Calculating the Inverse of a Matrix After the previous slightly complex definitions, the calculation of the inverse matrix is relatively simple.  Clearly, if the  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Determinants#_Determinants" determinant of A is zero, the inverse cannot be calculated and the matrix is said to be singular. 5 Application Solving Linear Equations One application of matrices is in solving linear equations (or simultaneous equations as they are often known). For example:  This can be written in terms of matrices:  or more generally  To solve this we simply need to pre-multiply both sides by the  HYPERLINK "http://www.morello.co.uk/matrixalgebra.htm" \l "_Calculating_the_Inverse#_Calculating_the_Inverse" inverse of A  In this case the answer is  Eigenvalues ( HYPERLINK "http://www.euclideanspace.com/maths/algebra/matrix/functions/eigenv/index.htm" http://www.euclideanspace.com/maths/algebra/matrix/functions/eigenv/index.htm) The eigenvalues of a matrix [M] are the values of  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET  such that: [M] {v} =  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET {v} where {v} = a vector this gives: |M -  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET I| = 0 where I = identity matrix this gives:  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/functions/eigenv/eigenvalue.gif" \* MERGEFORMATINET  so (m11-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) (m22-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) (m33-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) + m12 m23 m31 + m13 m21 m32 - (m11-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) m23 m32 - m12 m 21 (m33-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) - m13 (m22-  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET ) m31 = 0 the values of  INCLUDEPICTURE "http://www.euclideanspace.com/maths/algebra/matrix/lambda.gif" \* MERGEFORMATINET  are the eigenvalues of the matrix.  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