Matrix theory
From open-encyclopedia.com - the free encyclopedia.
In mathematics, matrix theory is a branch of mathematics which focuses on the study of matrices. Initially, a sub-branch of linear algebra, it soon grew to cover subjects related to graph theory, algebra, combinatorics and statistics as well.
| Contents |
History
The study of matrices is quite old. Latin squares and magic squares have been studied since prehistoric times.
Leibniz, one of the two founders of calculus, developed the theory of determinants in 1693 to aid in the solution of linear equations. Cramer developed the theory further, presenting Cramer's rule in 1750.
In 1800s, Gauss-Jordan elimination was invented.
It was J. J. Sylvester who first coined the term "matrix" in 1848. Cayley, Hamilton, Hermann Grassmann, Frobenius and von Neumann were among the famous mathematicians who had worked on matrix theory.
Elementary introduction
A matrix is a rectangular array of numbers. It can be identified with a linear transformation between two vector spaces. Therefore matrix theory is usually considered as a branch of linear algebra. The square matrices play a special role, because the nxn matrices for fixed n have many closure properties
In graph theory, each labelled graph corresponds to a unique non-negative matrix, the adjacency matrix. A permutation matrix is the matrix representation of a permutation; it is a square matrix with entries 0 and 1, with just one entry 1 in each row and each column. These types of matrices are used in combinatorics.
The ideas of stochastic matrix and doubly stochastic matrix are important tools to study stochastic processes, in statistics.
Positive-definite matrices occur in the search for maxima and minima of real-valued functions, when there are several variables.
It is also important to have a theory of matrices over arbitrary rings. In particular, matrices over polynomial rings are used in control theory.
Within pure mathematics, matrix rings can provide a rich field of counterexamples for mathematical conjectures, amongst other uses.
Some useful theorems
- Cayley-Hamilton theorem
- Jordan decomposition
- QR decomposition
- Schur triangulation
- Singular value decomposition
External links
fr:Théorie des matrices
