In Mathematical, the theory of the matrices is a branch of the Mathématiques which is interested in the study of the matrices. In the beginning, the theory of the matrices was regarded as a secondary branch of the Linear algebra, but increases for soon covering subjects relating to the Graph theory, with the Algèbre, the Combinatoire and the Statistiques.

The matrices are now used for multiple applications and are in particular used to represent the coefficients of the systems of linear equations or to represent the linear applications; in this last case the matrices play the same part as the coordinates of a vector for the linear applications.

History

The study of the matrices is completely old. The square square Latin and magic were studied for a very long time. Leibniz, one of the two founders of the analysis, developed the theory of the determinant S in 1693 to facilitate the resolution of the linear equations. Cramer looked further into this theory, by presenting the method of Cramer in 1750. In the Years 1800, the method of elimination of Gauss-Jordan was developed. It was James Sylvester which used for the first time the term “matrix” in 1850. Cayley, Hamilton, Hermann Grassmann, Frobenius and John von Neumann counts among the famous Mathématiciens which worked on the theory of the matrices.

In 1925, Werner Heisenberg redécouvre matrix algebra by melting a first formulation of what was going to become the quantum Mécanique. He is for this reason regarded as one of the fathers of quantum mechanics.

Elementary introduction

See also: Matrix (mathematics)

A matrix is a rectangular table of numbers. A matrix can be identified with a Linear application between two vector spaces of finished size. Thus the theory of the matrices is usually regarded as a branch of the Linear algebra. The square matrices play a particular part, because the whole of the matrices of order N ( N whole naturalness not no one given) has properties of “stability” of the operations.

The concepts of stochastic Matrix and doubly stochastic matrix are important tools to study the stochastic processes, in Probabilité and Statistique.

The positive definite matrices appear in the search for maximum and minimum of functions to actual values, and several variables.

It is also important to have a theory of the matrices with coefficients in a ring. In particular, the matrices with coefficients in the ring of the polynomials are used in Control theory.

In pure mathematics, the rings of matrices can provide a rich person field of counterexamples for mathematical conjectures.

Stamp and graph

In Graph theory, to any labelled graph corresponds the Matrice of adjacency . A Matrice of permutation is a matrix which represents a Permutation; square matrix whose coefficients are 0 or 1, with only one 1 in each line and each column. These matrices are used in Combinatoire.

In the Graph theory, one calls matrix of a graph the matrix indicating in the line I and the column J the number of edges connecting the top I to the top J . In a graph not directed, the matrix is symmetrical. The sum of the elements of a column makes it possible to determine the degree of a top. The matrix $M^n$ indicates in the line I and the column J the number of ways to N edges uniting the top I to the top J

Some theorems

• Theorem of Cayley-Hamilton

See too

• decomposition of Jordan

• elimination of Gauss-Jordan
• Decomposition LU
• Decomposition QR
• Lemma of Schur
• Decomposition in singular values

External bonds

• '' has off Brief History Linear Algebra and Matrix Theory '' (Short history of the linear algebra and the theory of the matrices)