Matrix Ring - Examples

Examples

• The set of all n×n matrices over an arbitrary ring R, denoted M_n(R). This is usually referred to as the "full ring of n by n matrices". These matrices represent endomorphisms of the free module Rn.

• The set of all upper (or set of all lower) triangular matrices over a ring.

• If R is any ring with unity, then the ring of endomorphisms of as a right R module is isomorphic to the ring of column finite matrices whose entries are indexed by, and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices whose rows each only have finitely many nonzero entries.

• If R is a normed ring, then the condition of row or column finiteness in the previous point can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. This idea can be used to represent operators on Hilbert spaces, for example.

• The intersection of the row and column finite matrix rings also forms a ring, which can be denoted by .

• The algebra M₂(R) of 2 × 2 real matrices is a simple example of a non-commutative associative algebra. Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: E₁₁E₂₁ = 0, hence it is not a division ring. Its invertible elements are nonsingular matrices and they form a group, the general linear group GL(2,R).

• If R is commutative, the matrix ring has a structure of a *-algebra over R, where the involution * on M_n(R) is the matrix transposition.

• Complex matrix algebras M_n(C) are, up to isomorphism, the only simple associative algebras over the field C of complex numbers. For n = 2, the matrix algebra M₂(C) plays an important role in the theory of angular momentum. It has an alternative basis given by the identity matrix and the three Pauli matrices. M₂(C) was the scene of early abstract algebra in the form of biquaternions.

• A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: σ(A,B)=tr(AB).