# 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.

## Some useful theorems

- Cayley-Hamilton theorem
- Jordan decomposition
- QR decomposition
- Schur triangulation
- Singular value decomposition