Die space

In mathematics, the matrix space or space of matrices is the vector space of the matrices of fixed size over a given body with the matrix addition and the scalar multiplication as inner and outer connections . The standard basis for the matrix space consists of the standard matrices in which exactly one entry is one and all other entries are zero . The dimension of the matrix space is equal to the product of the number of rows and columns in the matrix.

The matrix spaces have a fundamental meaning in linear algebra , since the space of the linear mappings between two finite-dimensional vector spaces is isomorphic (structurally equal) to a matrix space. According to this, after choosing a basis for the archetype and the target space, each linear mapping can be represented by a matrix and, conversely, each matrix corresponds to a linear mapping.

definition

Is a body as well as and natural numbers , so is ${\displaystyle K}$ ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle K^{m\times n}=\{(a_{ij})_{i=1,\ldots ,m,j=1,\ldots ,n}\mid a_{ij}\in K\}}$

the set of matrices of size with entries . A component-wise addition is now defined for matrices ${\displaystyle m\times n}$ ${\displaystyle K}$ ${\displaystyle A,B\in K^{m\times n}}$

${\displaystyle A+B=(a_{ij}+b_{ij})}$,

as well as a component-wise multiplication with a scalar by ${\displaystyle c\in K}$

${\displaystyle c\cdot A=(c\cdot a_{ij})}$.

In this way a vector space is obtained , which is called the matrix space or the space of matrices of size over the body . ${\displaystyle (K^{m\times n},+,\cdot )}$ ${\displaystyle m\times n}$ ${\displaystyle K}$

example

If one considers the space of the matrices of the size , then the matrix addition corresponds to exactly ${\displaystyle K^{2\times 2}}$ ${\displaystyle 2\times 2}$

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

and the scalar multiplication accordingly

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

As a result of the addition or scalar multiplication, one again obtains a matrix. ${\displaystyle (2\times 2)}$

properties

Neutral and inverse element

The neutral element in the matrix space is the zero matrix

${\displaystyle 0=(~0_{K}~)}$,

whose elements are all equal to the zero element of the body . The additively inverse element of a matrix is then the matrix ${\displaystyle 0_{K}}$ ${\displaystyle K}$ ${\displaystyle A=(a_{ij})}$

${\displaystyle -A=(-a_{ij})}$,

wherein for and , respectively, the additive inverse element to in is. ${\displaystyle -a_{ij}}$ ${\displaystyle i=1,\ldots ,m}$ ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle a_{ij}}$ ${\displaystyle K}$

Laws

The matrix space satisfies the axioms of a vector space . In addition to the existence of a neutral and inverse element, matrices and scalars apply ${\displaystyle A,B,C\in K^{m\times n}}$ ${\displaystyle c,d\in K}$

• the associative law , ${\displaystyle A+(B+C)=(A+B)+C}$
• the commutative law , ${\displaystyle A+B=B+A}$
• the mixed associative law , ${\displaystyle c\cdot (d\cdot A)=(c\cdot d)\cdot A}$
• the distributive laws and as well ${\displaystyle c\cdot (A+B)=c\cdot A+c\cdot B}$ ${\displaystyle (c+d)\cdot A=c\cdot A+d\cdot A}$
• the neutrality of one , being the one element of the body . ${\displaystyle 1_{K}\cdot A=A}$ ${\displaystyle 1_{K}}$ ${\displaystyle K}$

These laws follow directly from the associativity, commutativity and distributivity of addition and multiplication in the body by applying to each element of a matrix. ${\displaystyle K}$

Basis and dimension

The standard basis for the die space consists of the set of standard dies

${\displaystyle \{E_{ij}\mid i=1,\ldots ,m,j=1,\ldots ,n\}}$.

where the entry is one and all other entries are zero. Each matrix can thus be used as a linear combination ${\displaystyle (i,j)}$ ${\displaystyle A\in K^{m\times n}}$

${\displaystyle A=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}\cdot E_{ij}}$

of these basic matrices. The dimension of the matrix space is accordingly

${\displaystyle \operatorname {dim} (K^{m\times n})=m\cdot n}$,

it is therefore the product of the number of rows and the number of columns in the matrices of the space.

Isomorphism

The vector space of the matrices is isomorphic to the space of the linear mappings between two finite-dimensional vector spaces and over the same body , that is ${\displaystyle L(V,W)}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle K}$

${\displaystyle L(V,W)\cong K^{m\times n}}$,

where is the dimension of and the dimension of . Each linear mapping may namely the choice of a base for and for by ${\displaystyle m}$ ${\displaystyle W}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle f\in L(V,W)}$ ${\displaystyle \{v_{1},\ldots ,v_{n}\}}$ ${\displaystyle V}$ ${\displaystyle \{w_{1},\ldots ,w_{m}\}}$ ${\displaystyle W}$

${\displaystyle f(v_{j})=\sum _{i=1}^{m}a_{ij}w_{i}}$

for being represented. Thus, each such linear mapping can be uniquely described by a matrix , the so-called mapping matrix . Conversely, each matrix corresponds exactly to a linear mapping in this way . ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle (a_{ij})\in K^{m\times n}}$ ${\displaystyle L(V,W)}$

Extensions

The matrix space can be expanded, for example, by the following mathematical structures :

• Is a real or complex die space with a scalar product is provided, for example the Frobenius inner product , one obtains a scalar product . Since this space with respect to the induced by the scalar metric completely is, these are even a Hilbert space .

• If a real or complex matrix space is provided with a matrix norm , for example a natural matrix norm or the Frobenius norm , a normalized space is obtained . This space is then also complete with regard to the metric induced by the norm , i.e. a Banach space .

• If a matrix space is provided with a topology , a topological vector space is obtained , that is, the matrix addition and the scalar multiplication are continuous operations.

• If a space is square matrices in addition to the matrix addition and scalar multiplication with the matrix multiplication provided, you get a associative algebra .

See also

• Coordinate space , the vector space of coordinate vectors over a body
• Die ring, the ring of square dies over a ring
• General linear group , the group of regular matrices over a ring

literature

• Gerd Fischer : Linear Algebra: an introduction for first-year students . Springer, 2008, ISBN 3-8348-9574-1 .
• Michael Artin : Algebra . Springer, 1998, ISBN 3-7643-5938-2 .

Individual evidence

1. ↑ Fischer: Linear Algebra: An Introduction for New Students . S. 75 .
2. ^ Artin: Algebra . S. 125-127 .