Matrix ring

The matrix ring, matrix ring or ring of matrices in mathematics, the ring of square matrices of fixed size with entries from another, underlying ring. The additive and the multiplicative link in the die ring, the matrix addition and matrix multiplication. The neutral element in the matrix ring is the zero matrix and the identity element is the identity matrix. The matrix ring is Morita- equivalent to its underlying ring and therefore inherits many of its properties. However, the matrix ring is not commutative in general, even if the underlying ring should be commutative.

The matrix ring has in the ring theory of special importance, since each endomorphism ring of a free right module is isomorphic to a matrix ring of finite basis. Many rings can thus be implemented as a ring of a die ring. This procedure is called by analogy with the permutation representation of a group matrix representation of the ring.

• 4.1 determinant
• 4.2 Rank

• 5.1 subrings
• 5.2 units
• 5.3 ideals

Definition

Is a unitary ring, then forms the set of square matrices with entries from this ring

Together with the matrix addition and matrix multiplication as a two digit shortcuts again a unitary ring

The ring of matrices called matrix ring over or short. The matrix ring is also listed by, or.

Example

A simple example of a die ring, the amount of matrices to the matrix addition

And the matrix multiplication

The result is back one matrix. The addition and multiplication in the matrix ring and agents are usually represented by the same symbols.

Properties

Ring axioms

The amount of square matrices filled with the matrix addition and matrix multiplication, the ring axioms:

• It forms the matrix addition a commutative group after commutative group.
• It forms a semigroup of matrix multiplication due to the associativity of matrix multiplication.
• The distributive laws are valid due to the distributivity of matrix multiplication with the matrix addition.
• The neutral element with respect to addition in the matrix ring is the zero matrix

• The identity element in the matrix ring is the identity matrix

Zero divisor

Is the zero matrix in the matrix ring, an absorbent element, that is applicable for all matrices

The matrix ring is not zero divisors, because of not necessarily follow or. So true, for example

The matrix ring is therefore of no integrity ring. Accordingly, in matrix equations may also not be shortened because of does not necessarily follow.

Noncommutativity

The matrix ring is not commutative, even if it should be commutative, since it applies, for example,

The matrix ring is commutative if and only if is and is commutative.

The center of the die ring, so the amount of the elements that commute with any other is

Where the center is.

Isomorphisms

The matrix ring is isomorphic to the ring of endomorphisms ( self-images ) of the free right module, so

The component- wise addition of pictures here corresponds to the matrix addition and the consecutive execution of mappings of matrix multiplication. The zero matrix corresponds to the zero mapping and the fuel matrix, the identity map.

A unitary ring is isomorphic to the matrix ring when it comes, a lot of elements, so that

As well as

Apply and if the centralizer of these elements in is isomorphic to.

Parameters

Determinant

Is commutative, then the determinant of a matrix is defined as the normalized alternating multilinear form. The determinant of a matrix can then use the Leibniz formula

Are determined, the sum over all permutations of the symmetric group, and passes the degree indicates the sign of a permutation. For the determinant of the product of two matrices of the determinants of product rate applies

Rank

The column rank of a matrix is defined as the maximum number of linearly independent column vectors in the free right module. According to the row rank of a matrix is ​​the maximum number of linearly independent row vectors. Is commutative, then vote column rank and row rank match and it is called the rank of the matrix,

Applies. Then for the rank of the product of two matrices

Substructures

Subrings

The matrices with entries from a Untering also form a subring of the matrix ring. However, matrix rings have other sub-rings. For example, structural rings are formed by:

• The set of diagonal matrices; this subring is commutative if is commutative
• The set of ( strictly ) upper or ( strictly ) lower triangular matrices
• The amount of Blockdiagonalmatrizen or Blockdreiecksmatrizen
• The amount of the matrix in which certain rows or columns have only zero entries

Many rings can be realized as a ring of a die ring. This procedure is called by analogy with the permutation representation of a group matrix representation of the ring. This sub- rings are sometimes referred to as matrix rings, and then called the matrix ring to better differentiate full matrix ring.

Units

The unit group in the matrix ring is the general linear group consisting of the regular matrices. For the inverse of the product of two normal matrices

A matrix is ​​invertible if its columns form a basis of the free right module. Is commutative, then there exists at all a matrix adjoint, so

Applies. In this case, the invertibility of a matrix is equivalent to the invertibility of its determinant.

Ideal

The ideals of the matrix ring are just given by, where is an ideal of. The factor rings of the cavity ring be so by

Characterized.

Specifically, is a body or skew field, then the matrix ring is simple, that is, he has only the zero ring and the whole ring as trivial ideals. By the theorem of Artin - Wedderburn every semisimple ring is isomorphic to a finite direct product of matrix rings over skew fields. With the component-wise scalar multiplication of the matrix ring forms an associative algebra.