Matrix theory

Matrix theory is a branch of mathematics which focuses on the study of matrices. Initially a sub-branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics, and statistics as well.

Overview

A matrix is a rectangular array of numbers. For an elementary article on matrices, their basic properties, and history, see the article matrix (mathematics).

A matrix 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 n×n matrices for fixed n have many closure properties.

In graph theory, each labeled 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.

Useful theorems

• Cayley–Hamilton theorem
• Jordan decomposition
• QR decomposition
• Schur triangulation
• Singular value decomposition

References

• Beezer, Rob, A First Course in Linear Algebra, licensed under GFDL.
• Jim Hefferon: Linear Algebra (Online textbook)

External links

• A Brief History of Linear Algebra and Matrix Theory