matriisialgebra

Matriisialgebra is the algebra of n by n matrices over a field F, denoted M_n(F). It consists of all n×n matrices with entries in F, equipped with ordinary matrix addition and multiplication. As an algebra, it is unital, having the identity matrix I_n, and it has dimension n^2 as a vector space over F.

A natural basis for Matriisialgebra is provided by the matrix units E_ij, where E_ij has a 1 in the (i, j) position and 0 elsewhere. The operations satisfy associativity and distributivity, and matrix multiplication is in general noncommutative for n greater than 1.

Key structural properties include that the center of M_n(F) consists precisely of the scalar matrices λI_n, with λ in F. The algebra is simple, meaning it has no nontrivial two-sided ideals. There is a canonical isomorphism M_n(F) ≅ End_F(F^n), the endomorphism algebra of the n-dimensional vector space F^n, with the action given by left multiplication.

Representations of Matriisialgebra are closely tied to the defining module F^n: under left multiplication, F^n becomes a simple M_n(F)-module, and up to isomorphism this module is unique. Invariants such as the trace and determinant are defined for elements of M_n(F) and play central roles in many applications.

Variants include forming matrix algebras over different fields (real, complex, finite fields) or over rings. Matriisialgebra plays a fundamental role in linear algebra, representation theory, and mathematical physics, where it models linear transformations and operator algebras.