Matrix ring

"Matrix algebra" redirects here. For the algebraic theory of matrices, see Matrix (mathematics) and Linear algebra.

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of n × n matrices with entries from R is a matrix ring denoted Mn(R), as well as some subsets of infinite matrices which form infinite matrix rings. Any subring of a matrix ring is a matrix ring.

When R is a commutative ring, the matrix ring Mn(R) is an associative algebra, and may be called a matrix algebra. For this case, if M is a matrix and r is in R, then the matrix Mr is the matrix M with each of its entries multiplied by r.

This article assumes that R is an associative ring with a unit 1 ≠ 0, although matrix rings can be formed over rings without unity.

Examples

Structure

Properties


  \begin{bmatrix}
    1 & 0 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}\,

and   \begin{bmatrix}
    1 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}\,
. This example is easily generalized to n×n matrices.

  \begin{bmatrix}
    0 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    0 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    0 & 0 \\
    0 & 0
  \end{bmatrix}\,
.

Diagonal subring

Let D be the set of diagonal matrices in the matrix ring Mn(R), that is the set of the matrices such that every nonzero entry, if any, is on the main diagonal. Then D is closed under matrix addition and matrix multiplication, and contains the identity matrix, so it is a subalgebra of Mn(R).

As an algebra over R, D is isomorphic to the direct product of n copies of R. It is a free R-module of dimension n. The idempotent elements of D are the diagonal matrices such that the diagonal entries are themselves idempotent.

Two dimensional diagonal subrings

When R is the field of real numbers, then the diagonal subring of M2(R), is isomorphic to split-complex numbers. When R is the field of complex numbers, then the diagonal subring is isomorphic to bicomplex numbers. When R = ℍ, the division ring of quaternions, then the diagonal subring is isomorphic to the ring of split-biquaternions, presented in 1873 by William K. Clifford.

Matrix semiring

In fact, R only needs to be a semiring for Mn(R) to be defined. In this case, Mn(R) is a semiring. If R={0,1} with 1+1=1, then Mn(R) is the semiring of binary relations on an n-element set with union as addition and composition as multiplication.

See also

References

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