Determinant (mathematics)
See also: Determinant
In Mathematical, initially introduced in Algebra to determine the number of solutions of a System of linear equations, the determinant appears a very powerful tool in many fields (study of endomorphism, search for eigenvalues, differential Calculus). Thus one defines the determinant of a system of equations, the determinant of a endomorphism or the determinant of a system of vectors.
As for many operations, the determinant can be defined by a collection of properties (Axiome S) which one summarizes by the term “forms N-linear alternate”. This definition makes it possible to make a complete theoretical study and to still widen its fields of application of it. But the determinant can be also conceived like a generalization with the space of dimension N of the concept of Surface or directed Volume. This aspect, often neglected, is a practical and lighting approach properties of the determinant.
History of the determinants
The determinants were introduced in Occident as from the 16th century, that is to say well before the matrices, which appear only at the 19th century. It is advisable to recall that the Chinese were the first to use tables of numbers and to apply a now known algorithm under the name of process of elimination of Gauss-Jordan.
The first calculations of determinants
In its original direction, the determinant determines the unicity of the solution of a Système of linear equations. It was introduced into the case of size 2 by Cardan in 1545 in its Ars Magna , in the form of a rule for the resolution of systems of two equations to two unknown factors. This first formula bears the name of controlled modo .
The appearance of the determinants of higher size requires even more than one hundred years. Curiously Japanese Kowa Seki and German Leibniz gave the first examples of them almost simultaneously.
Leibniz studies many systems of linear equations. In the absence of matric notation, it represents the unknown coefficients by a couple of indices: it notes thus ij for ai, j . In 1678, it is interested in a system of three equations and three unknown factors and gives, on this example, the formula of development following a column. The same year, he writes a determinant of size 4, correct except for the signs. Leibniz does not publish this work, which seems to be forgotten before the results are redécouverts later independently about fifty years.
At the same period, Kowa Seki publishes a manuscript on the determinants, where it finds a formulation general difficult to interpret. It seems to give correct formulas for determinants of size 3 and 4, and again of the erroneous signs for the determinants of higher size. The discovery will remain without a future, because of the cut of Japan with the outside world.
Determinants of unspecified size
In 1748, a posthumous treaty of algebra of MacLaurin revival the theory of the determinants, with the correct writing of the solution of a system of four equations and four unknown factors.
In 1750, Cramer formula the rules which make it possible to solve a system of N equations and unknown N , but without giving the demonstration of it. The methods of calculating of the determinants are then delicate, since founded on the concept of signature of a permutation.
The mathematicians seize this new object, with articles of Bézout in 1764, of Vandermonde in 1771 (surprisingly not giving the calculation of the determinant of the current Matrice of Vandermonde). In 1772, Laplace establishes the formulas of recurrence bearing its name. The following year, Lagrange discovers the bond between the calculation of the determinants and volumes.
Gauss uses for the first time the word “determining”, in the Disquisitiones arithmeticae in 1801. It employs it for what we describe today as Discriminant of a Quadrique and which is a particular case of the modern determinant. It is also close obtaining the theorem on the determinant of a product.
Installation of the modern concept of determinant
Cauchy employs the first the word determining in its modern direction. One can thus read in his article of synthesis of more than four twenty pages on the question:
Mr. Gauss was useful himself about it with advantage in his analytical Research to discover the general properties of the forms of the second degree, i.e. the Polynôme S of the second degree with two or several variables, and it indicated these same functions under the name of determinants. I will preserve this denomination which provides a means easy to state the results; I will observe only also sometimes that one gives to the functions in question the name of resultants to two or with several letters. Thus the two following expressions, determinant and resulting, will have to be looked like synonyms.
It represents a synthesis of former knowledge, as of the news proposal like the fact that the transposed Application does not modify the determinant as well as the formula of the determinant of a product. Binet also proposes a demonstration this same year. Later, Cauchy provides the foundations of the study of the Réduction of endomorphism S.
By publishing its three treaties on the determinants in 1841 in the Newspaper of Crelle, Jacobi gives a true notoriety to the concept
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