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.
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.
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