Die ring

The matrix ring , matrix ring or ring of the matrices is in the mathematics of the ring of the square matrices of fixed size with entries from a further underlying ring. The additive and the multiplicative link in the matrix ring are the matrix addition and the matrix multiplication . The neutral element in the matrix ring is the zero matrix and the identity , the identity matrix . The matrix ring is Morita equivalent to its underlying ring and therefore inherits many of its properties. However, the die ring is generally non- commutative , even if the underlying ring should be commutative.

The matrix ring has a special meaning in ring theory , since every endomorphism ring of a free module with a finite base is isomorphic to a matrix ring . Many rings can thus be implemented as the lower ring of a die ring . In analogy to the permutation representation of a group, this procedure is called a matrix representation of the ring.

definition

If a unitary ring , then the set of square matrices forms with entries from this ring ${\displaystyle (R,+,\cdot )}$

${\displaystyle R^{n\times n}=\{(r_{ij})\mid r_{ij}\in R~\mathrm {f{\ddot {u}}r} ~i,j=1,\dotsc ,n\}}$

together with the matrix addition and the matrix multiplication as two-digit connections, in turn, a unitary ring

${\displaystyle (R^{n\times n},+,\cdot )}$,

the ring of dies is called the die ring for short. The addition and multiplication in the matrix ring and in the ring on which it is based are usually represented by the same symbols. The die ring is also noted as , or . ${\displaystyle R}$ ${\displaystyle R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle M_{n}(R)}$ ${\displaystyle \operatorname {Mat} (n,R)}$ ${\displaystyle R_{n}}$

example

A simple example of a matrix ring is the set of matrices with the matrix addition ${\displaystyle (2\times 2)}$

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\end{pmatrix}}}$

and the matrix multiplication

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}\cdot {\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}a_{11}\cdot b_{11}+a_{12}\cdot b_{21}&a_{11}\cdot b_{12}+a_{12}\cdot b_{22}\\a_{21}\cdot b_{11}+a_{22}\cdot b_{21}&a_{21}\cdot b_{12}+a_{22}\cdot b_{22}\end{pmatrix}}}$.

The result is another matrix. ${\displaystyle (2\times 2)}$

properties

Ring axioms

The set of square matrices satisfies the ring axioms with the matrix addition and the matrix multiplication :

• Together with the matrix addition , it forms a commutative group after there is a commutative group. ${\displaystyle (R,+)}$
• Together with the matrix multiplication, it forms a semigroup due to the associativity of the matrix multiplication .
• The distributive laws apply due to the distributivity of the matrix multiplication with the matrix addition .
• The neutral element with regard to the addition in the matrix ring is the zero matrix

${\displaystyle 0={\begin{pmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{pmatrix}}}$,

where is the neutral element of . ${\displaystyle 0}$ ${\displaystyle R}$

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

${\displaystyle I={\begin{pmatrix}1&0&\cdots &0\\0&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&1\end{pmatrix}}}$,

where the element is of. In order to rule out trivial cases, it is assumed below. ${\displaystyle 1}$ ${\displaystyle R}$ ${\displaystyle 1\neq 0}$

Zero divisor

The zero matrix is matrix ring an absorbent element , that is, for all matrices applies ${\displaystyle R^{n\times n}}$ ${\displaystyle A\in R^{n\times n}}$

${\displaystyle A\cdot 0=0\cdot A=0}$.

The die ring is not free of zero divisors because it does not necessarily follow or . For example ${\displaystyle n>1}$ ${\displaystyle A\cdot B=0}$ ${\displaystyle A=0}$ ${\displaystyle B=0}$

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}$.

The die ring is therefore not an integrity ring . Correspondingly, matrix equations must not be shortened, because it does not necessarily follow . ${\displaystyle n>1}$ ${\displaystyle A\cdot B=A\cdot C}$ ${\displaystyle B=C}$

Non-commutativity

The matrix ring is for non- commutative , even if it should be commutative, because it holds for example ${\displaystyle R^{n\times n}}$ ${\displaystyle n>1}$ ${\displaystyle R}$

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}0&0\\1&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\neq {\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\cdot {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$.

The matrix ring is commutative if and only if is and is commutative. ${\displaystyle R^{n\times n}}$ ${\displaystyle n=1}$ ${\displaystyle R}$

The center of the die ring, i.e. the set of elements that commute with all the others, is

${\displaystyle Z(R^{n\times n})=\{z\cdot I\mid z\in Z(R)\}}$,

where is the center of . ${\displaystyle Z(R)}$ ${\displaystyle R}$

Isomorphies

The matrix ring is isomorphic to the ring of endomorphisms (self-images) of the free module , i.e. ${\displaystyle R^{n\times n}}$ ${\displaystyle R^{n}}$

${\displaystyle R^{n\times n}\cong \operatorname {End} (R^{n})}$.

The component-wise addition of images corresponds to the matrix addition and the sequential execution of images corresponds to the matrix multiplication. The zero matrix corresponds to the zero mapping and the one matrix corresponds to the identical mapping .

A unitary ring is isomorphic to the die ring , if there are a lot of elements , , are such that ${\displaystyle S}$ ${\displaystyle R^{n\times n}}$ ${\displaystyle n^{2}}$ ${\displaystyle e_{ij}}$ ${\displaystyle i,j=1,\dotsc ,n}$

${\displaystyle e_{ij}e_{kl}=\delta _{jk}e_{il}}$

such as

${\displaystyle e_{11}e_{11}+e_{22}e_{22}+\dotsb +e_{nn}e_{nn}=1}$

apply and if the centralizer of these elements is in isomorphic to . ${\displaystyle S}$ ${\displaystyle R}$

Parameters

Determinant

→ Main article : determinant

If commutative, then the determinant of a matrix is ​​defined as a normalized, alternating multilinear form. The determinant of a matrix can then be via the Leibniz formula ${\displaystyle R}$ ${\displaystyle R^{n\times n}\to R}$

${\displaystyle \det A=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn} (\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}$

can be determined, the sum of all permutations of the symmetrical group running from degree and denoting the sign of a permutation. The determinant product theorem applies to the determinant of the product of two matrices ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle \operatorname {sgn} }$

${\displaystyle \det(A\cdot B)=\det A\cdot \det B}$.

rank

→ Main article : Rank (mathematics)

The column rank of a matrix is ​​defined as the maximum number of linearly independent column vectors in the free module . Correspondingly, the row rank of a matrix is ​​the maximum number of linearly independent row vectors. If commutative, then column rank and row rank match and one speaks of the rank of the matrix, where ${\displaystyle R^{n}}$ ${\displaystyle R}$

${\displaystyle \operatorname {rang} A\leq n}$

applies. The following then applies to the rank of the product of two matrices

${\displaystyle \operatorname {rang} (A\cdot B)\leq \min\{\operatorname {rang} A,\operatorname {rang} B\}}$.

Substructures

Sub-rings

The square matrices with entries from a Untering of also forming a bottom ring in the die ring . However, die rings have further sub-rings. For example, structural sub-rings are formed by: ${\displaystyle U}$ ${\displaystyle R}$ ${\displaystyle U^{n\times n}}$ ${\displaystyle R^{n\times n}}$

• the set of diagonal matrices ; this subring is commutative if is commutative ${\displaystyle R}$
• the set of (strictly) upper or (strictly) lower triangular matrices
• the set of block diagonal matrices or block triangle matrices
• the set of matrices in which certain columns or rows only have zero entries

Many rings can be implemented as the lower ring of a die ring. In analogy to the permutation representation of a group, this procedure is called a matrix representation of the ring. These sub-rings are sometimes also referred to as die rings and the die ring is then called a full die ring for better differentiation. ${\displaystyle R^{n\times n}}$

units

The unit group in the matrix ring is the general linear group consisting of the regular matrices . The inverse of the product of two regular matrices holds ${\displaystyle R^{n\times n}}$ ${\displaystyle \operatorname {GL} (n,R)}$ ${\displaystyle A,B\in R^{n\times n}}$

${\displaystyle (A\cdot B)^{-1}=B^{-1}\cdot A^{-1}}$.

A matrix is ​​invertible if and only if its columns form the basis of the free module . If commutative, then there is an adjunct for every matrix such that ${\displaystyle R^{n}}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle \operatorname {adj} A}$

${\displaystyle A\cdot \operatorname {adj} A=\operatorname {adj} A\cdot A=\det A\cdot I}$

applies. In this case the invertibility of a matrix is ​​equivalent to the invertibility of its determinant in . ${\displaystyle \det A}$ ${\displaystyle R}$

Ideals

The ideals of the die ring are going through , where, with an ideal of is. The factor rings of the matrix ring are thus through ${\displaystyle R^{n\times n}}$ ${\displaystyle J^{n\times n}}$ ${\displaystyle J}$ ${\displaystyle R}$

${\displaystyle R^{n\times n}/J^{n\times n}\cong (R/J)^{n\times n}}$

characterized.

Matrix algebra

If a body or an oblique body is special , then the matrix ring is simple , that is, it has only the zero ring and the whole ring as trivial ideals. According to Artin-Wedderburn's theorem , every semisimple ring is isomorphic to a finite direct product of matrix rings over oblique fields. With the component-wise scalar multiplication , the matrix ring forms an associative algebra . ${\displaystyle R=K}$ ${\displaystyle K^{n\times n}}$ ${\displaystyle \{0\}}$ ${\displaystyle K^{n\times n}}$ ${\displaystyle K^{n\times n}}$

See also

• Matrix space , the vector space of the matrices over a body
• Matrix representation of quaternions

literature

• Michael Artin : Algebra . Springer, 1998, ISBN 3-7643-5938-2 .
• Paul Cohn : An Introduction to Ring Theory . Springer, 2000, ISBN 1-85233-206-9 .
• Serge Lang : Algebra . 3. Edition. Springer, 2002, ISBN 0-387-95385-X .

Individual evidence

1. ↑ ^{a b c d e} D.A. Suprunenko: Matrix ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
2. ^ OA Ivanova: Rank . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Web links

• DA Suprunenko: Matrix ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• CWoo: Matrix ring . In: PlanetMath . (English)