Standard matrix

A standard matrix , standard unit matrix or matrix unit is in mathematics a matrix in which one entry is one and all other entries are zero. Every standard matrix can be represented as a dyadic product of canonical unit vectors . The set of standard matrices forms the standard basis for the matrix space . Among other things, they are used to define elementary matrices that are used in the Gaussian elimination process .

definition

If a ring with zero element and one element , then the standard matrix is the matrix with the entries ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle E_{ij}=(e_{kl})\in R^{m\times n}}$

${\displaystyle e_{kl}={\begin{cases}1&{\text{für }}i=k{\text{ und }}j=l\\0&{\text{sonst}}\end{cases}}}$

for and . In the standard matrix, the entry at this point is therefore equal to one and all other entries are equal to zero. A standard matrix is ​​also known as a standard identity matrix or matrix unit and is occasionally notated with instead of . ${\displaystyle k=1,\ldots ,m}$ ${\displaystyle l=1,\ldots ,n}$ ${\displaystyle E_{ij}}$ ${\displaystyle (i,j)}$ ${\displaystyle e_{ij}}$ ${\displaystyle E_{ij}}$

Examples

If the field of the real numbers and denotes and the numbers zero and one , then examples of standard matrices are : ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 3\times 3}$

${\displaystyle E_{11}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},E_{23}={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}}}$

properties

Representations

Every standard matrix can be represented as the dyadic product of the two canonical unit vectors and , that is ${\displaystyle E_{ij}\in R^{m\times n}}$ ${\displaystyle e_{i}\in R^{m}}$ ${\displaystyle e_{j}\in R^{n}}$

${\displaystyle E_{ij}=e_{i}\otimes e_{j}=e_{i}\cdot (e_{j})^{T}}$,

where the transposed vector is to. With the help of the Kronecker delta , a standard matrix can also be passed through ${\displaystyle e^{T}}$ ${\displaystyle e}$

${\displaystyle E_{ij}=(\delta _{ik}\,\delta _{jl})_{k=1,\ldots ,m \atop l=1,\ldots ,n}=(\delta _{(i,j),(k,l)})_{k=1,\ldots ,m \atop l=1,\ldots ,n}}$

note.

symmetry

The following applies to the transpose of a standard matrix ${\displaystyle E_{ij}\in R^{m\times n}}$

${\displaystyle (E_{ij})^{T}=E_{ji}}$.

This means that only the standard matrices are symmetrical . ${\displaystyle E_{ii}\in R^{n\times n}}$

product

For the product of two standard matrices and applies ${\displaystyle E_{ij}\in R^{m\times n}}$ ${\displaystyle E_{kl}\in R^{n\times p}}$

${\displaystyle E_{ij}\cdot E_{kl}={\begin{cases}E_{il}&{\text{falls }}j=k\\0&{\text{sonst,}}\end{cases}}}$

where the zero matrix is the size . ${\displaystyle 0}$ ${\displaystyle m\times p}$

Parameters

The following applies to the rank of a standard matrix

${\displaystyle \operatorname {rang} (E_{ij})=1}$.

The same applies to the determinant and the trace of a square standard matrix ${\displaystyle m\times m}$

${\displaystyle \operatorname {det} (E_{ij})={\begin{cases}1&{\text{falls }}m=1\\0&{\text{sonst}}\end{cases}}}$   and   . ${\displaystyle \operatorname {spur} (E_{ij})=\delta _{ij}}$

The characteristic polynomial of a square standard matrix over a body is given by ${\displaystyle E_{ij}\in K^{n\times n}}$ ${\displaystyle K}$

${\displaystyle \chi (\lambda )={\begin{cases}\lambda ^{n-1}(\lambda -1)&{\text{falls }}i=j\\\lambda ^{n}&{\text{sonst.}}\end{cases}}}$

In the case is therefore the only eigenvalue . For there is also the eigenvalue with simple multiplicity and the associated eigenvector . ${\displaystyle i\neq j}$ ${\displaystyle 0}$ ${\displaystyle i=j}$ ${\displaystyle 1}$ ${\displaystyle e_{i}}$

use

Matrix entries

With the help of standard matrices, individual matrix entries can also be displayed as a trace. Is , then applies ${\displaystyle E_{ji}\in R^{n\times m}}$ ${\displaystyle A\in R^{m\times n}}$

${\displaystyle (A)_{ij}=\operatorname {spur} (AE_{ji})=\operatorname {spur} (E_{ji}A)}$.

For the product of two matrices and applies accordingly ${\displaystyle A\in R^{m\times p}}$ ${\displaystyle B\in R^{p\times n}}$

${\displaystyle (AB)_{ij}=\operatorname {spur} (BE_{ji}A)}$.

Standard base

The set of standard matrices over a given field forms the standard basis for the vector space of the matrices. Each matrix can thus be defined as a linear combination of standard matrices ${\displaystyle \{E_{ij}\in K^{m\times n}\mid i=1,\ldots ,m,j=1,\ldots ,n\}}$ ${\displaystyle A\in K^{m\times n}}$

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

with represent. Thus, the four standard dies form , , and the standard basis of the area of matrices and are obtained for example ${\displaystyle a_{ij}\in K}$ ${\displaystyle E_{11}}$ ${\displaystyle E_{12}}$ ${\displaystyle E_{21}}$ ${\displaystyle E_{22}}$ ${\displaystyle (2\times 2)}$

${\displaystyle {\begin{pmatrix}2&4\\3&1\end{pmatrix}}={\begin{pmatrix}2&0\\0&0\end{pmatrix}}+{\begin{pmatrix}0&4\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\3&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&1\end{pmatrix}}=2E_{11}+4E_{12}+3E_{21}+1E_{22}}$.

Elementary matrices

Standard matrices are also used to represent the three types of elementary matrices of the shape

${\displaystyle {\begin{aligned}R_{ij}(\alpha )&=I+\alpha E_{ij}\\S_{i}(\gamma )&=I+(\gamma -1)E_{ii}\\T_{i,j}&=I-E_{ii}-E_{jj}+E_{ij}+E_{ji}\end{aligned}}}$

with as the identity matrix and used. By multiplying from the left with such an elementary matrix, row operations, scalings and transpositions are carried out on a given matrix. These elementary matrices are used to describe the Gaussian elimination method for solving systems of linear equations . ${\displaystyle I}$ ${\displaystyle \alpha ,\gamma \in K}$

literature

• Tilo Arens, Frank Hettlich, Christian Karpfinger, Ulrich Kockelkorn, Klaus Lichtenegger, Hellmuth Stachel : Mathematics . 2nd Edition. Spektrum Akademischer Verlag, 2011, ISBN 3-8274-2347-3 .
• Michael Artin : Algebra . Springer, 1998, ISBN 3-7643-5938-2 .
• Christian Voigt, Jürgen Adamy: Collection of formulas for matrix calculation . Oldenbourg, 2007, ISBN 3-486-58350-6 .

Individual evidence

1. ^ Voigt, Adamy: Collection of formulas for matrix calculation . S. 8 .
2. ^ Arens et al: Mathematics . S. 508 .
3. ^ Artin: Algebra . S. 11 .