Elementary matrix

Under an elementary matrix or elimination matrix is understood in linear algebra , a square matrix , which either by the change of a single item or by swapping two lines of a - unit matrix is different. ${\displaystyle n\times n}$ ${\displaystyle I_{n}}$

The matrix multiplication with elementary matrices leads to the so-called elementary row and column transformations . These matrix transformations include adding times one row to another, swapping two rows, and multiplying a single row by a non-zero value . Multiplying a matrix from the left by an elementary matrix corresponds to an elementary line transformation of the matrix . Elementary matrices can also be multiplied from the right of a matrix and then correspond to elementary column transformations of . ${\displaystyle \alpha }$ ${\displaystyle \gamma }$ ${\displaystyle n\times p}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$

The elementary matrices are the basis for the Gaussian algorithm . With them, a linear system of equations , which has been converted into a matrix, can be stepped in order to then read off the solution of the system according to special rules.

Types of elementary matrices

In the following we assume a field , an - identity matrix and a - standard matrix , i.e. H. a matrix of zero elements, with the exception that there is a one element at the point , where the row index and the column index of the matrices are used. ${\displaystyle K}$ ${\displaystyle I_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle E_{i,j}}$ ${\displaystyle n\times n}$ ${\displaystyle (i,j)}$ ${\displaystyle i}$ ${\displaystyle j}$

There are three types of elementary matrices:

Type 1

This matrix has only one elements in its main diagonal , otherwise only zero elements, with the exception of the place where the value is, where must be - i.e. H. the value must not be on the main diagonal. ${\displaystyle (i,j)}$ ${\displaystyle \alpha \in K}$ ${\displaystyle i\neq j}$ ${\displaystyle \alpha }$

This is generated by

${\displaystyle I_{n}+\alpha \cdot E_{i,j}}$, where and is. ${\displaystyle \alpha \in K}$ ${\displaystyle i\neq j}$

For the abbreviation we write

${\displaystyle R_{i,j}(\alpha )=I_{n}+\alpha \cdot E_{i,j};}$

please note, however, that this is not a standard notation.

So it is explicitly true

${\displaystyle R_{i,j}(\alpha )={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\\\end{pmatrix}}+\alpha \cdot {\begin{pmatrix}0&0&0&\cdots &0\\0&0&0&1&0\\0&0&0&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &0\\\end{pmatrix}}={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\alpha &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\\\end{pmatrix}}}$,

where is in the -th row and -th column. ${\displaystyle \alpha }$ ${\displaystyle i}$ ${\displaystyle j}$

Examples

${\displaystyle R_{2,1}(-7)={\begin{pmatrix}1&0&0\\-7&1&0\\0&0&1\\\end{pmatrix}}}$

${\displaystyle R_{1,3}(-3)={\begin{pmatrix}1&0&-3&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}}$

Type 2

This matrix corresponds to an identity matrix in which the -th and -th lines have been swapped (of course ). In the main diagonal of , the single element is counted away at the places and (to get zero) and the single element is added again at the places and . This type is therefore the permutation matrix of a transposition . ${\displaystyle I_{n}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle i\neq j}$ ${\displaystyle I_{n}}$ ${\displaystyle (i,i)}$ ${\displaystyle (j,j)}$ ${\displaystyle (i,j)}$ ${\displaystyle (j,i)}$

The following matrix operations do this:

${\displaystyle I_{n}-E_{i,i}-E_{j,j}+E_{i,j}+E_{j,i}}$, For ${\displaystyle i\neq j}$

For abbreviation, we define type 2 here as

${\displaystyle T_{i,j}=I_{n}-E_{i,i}-E_{j,j}+E_{i,j}+E_{j,i}}$

The operations generally look like this:

${\displaystyle {\begin{pmatrix}1&&&&\\&1&&&\\&&\ddots &&\\&&&1&\\&&&&1\\\end{pmatrix}}-{\begin{pmatrix}&&&&\\&1_{(i,i)}&&&\\&&&&\\&&&&\\&&&&\\\end{pmatrix}}-{\begin{pmatrix}&&&&\\&&&&\\&&&&\\&&&1_{(j,j)}&\\&&&&\\\end{pmatrix}}+{\begin{pmatrix}&&&&\\&&&1_{(i,j)}&\\&&&&\\&&&&\\&&&&\\\end{pmatrix}}+{\begin{pmatrix}&&&&\\&&&&\\&&&&\\&1_{(j,i)}&&&\\&&&&\\\end{pmatrix}}=}$

${\displaystyle ={\begin{pmatrix}1&&&&\\&0_{(i,i)}&\cdots &1_{(i,j)}&\\&\vdots &1&\vdots &\\&1_{(j,i)}&\cdots &0_{(j,j)}&\\&&&&1\\\end{pmatrix}}}$

The following example shows how the -th and -th lines are swapped: ${\displaystyle i}$ ${\displaystyle j}$

example

${\displaystyle T_{1,2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}-{\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\\\end{pmatrix}}-{\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\\\end{pmatrix}}+{\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}}+{\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\\\end{pmatrix}}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\\\end{pmatrix}}}$

Is analog

${\displaystyle T_{2,4}={\begin{pmatrix}1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0\\\end{pmatrix}}}$

Type 3

The main diagonal of this matrix consists of single elements, except for the place where the value is inserted, which must be non-zero. Outside the main diagonal there are only zero elements. ${\displaystyle (i,i)}$ ${\displaystyle \gamma \in K}$

This is achieved via

${\displaystyle I_{n}+(\gamma -1)\cdot E_{i,i}}$, with and ${\displaystyle \gamma \in K}$ ${\displaystyle \gamma \neq 0}$

(At this point , add 1 and subtract 1.) ${\displaystyle (i,i)}$ ${\displaystyle \gamma }$

For abbreviation, type 3 should be used here as

${\displaystyle S_{i}(\gamma )=I_{n}+(\gamma -1)\cdot E_{i,i}}$

To be defined. Again, it is not a standard notation.

Operations performed:

${\displaystyle {\begin{pmatrix}1&&&&\\&1&&&\\&&\ddots &&\\&&&1&\\&&&&1\\\end{pmatrix}}+(\gamma -1)\cdot {\begin{pmatrix}&&&&&\\&1_{(i,i)}&&&&\\&&&&&\\&&&&&\\&&&&&\\\end{pmatrix}}={\begin{pmatrix}1&&&&\\&\gamma _{(i,i)}&&&\\&&\ddots &&\\&&&1\\&&&&1\\\end{pmatrix}}}$

Examples

${\displaystyle S_{2}(8)={\begin{pmatrix}1&0&0\\0&8&0\\0&0&1\\\end{pmatrix}}}$

${\displaystyle S_{3}(17)={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&17&0\\0&0&0&1\\\end{pmatrix}}}$

Influence of the elementary matrices on other matrices

Let A be a matrix and , and matrices of type 1, type 2 and type 3 , respectively. ${\displaystyle m\times n}$ ${\displaystyle R_{i,j}(\alpha )}$ ${\displaystyle T_{i,j}}$ ${\displaystyle S_{i}(\gamma )}$

Multiplication from the left gives line transformations:

• ${\displaystyle S_{i}(\gamma )\cdot A}$multiplies the i-th line of A by the value , whereby the remaining lines remain unchanged (EZU I) ${\displaystyle \gamma }$
• ${\displaystyle R_{i,j}(\alpha )\cdot A}$adds the -fold of the j-th row of A to the i-th row of A. (EZU II) ${\displaystyle \alpha }$
• ${\displaystyle T_{i,j}\cdot A}$ swaps the i-th row of A with the j-th row of A. (EZU III)

Multiplication from the right gives column transformations:

• ${\displaystyle A\cdot S_{i}(\gamma )}$multiplies the i-th column of A by the value , leaving the remaining columns unchanged. (ESU I) ${\displaystyle \gamma }$
• ${\displaystyle A\cdot R_{i,j}(\alpha )}$adds the -fold of the i-th column of A to the j-th column of A. (ESU II) Note the interchanged meaning of i and j in contrast to the line conversion. ${\displaystyle \alpha }$
• ${\displaystyle A\cdot T_{i,j}}$ swaps the i-th column of A with the j-th column of A. (ESU III)

See also matrix multiplication . These properties are important for solution methods of matrix calculations, such as the Gauss-Jordan algorithm .

Reminder: In order to construct the appropriate elementary matrix for one of the above-mentioned transformations, the corresponding transformation must be applied to the identity matrix . For example, to obtain the elementary matrix that swaps the first and second rows of a matrix, the first and second rows of the identity matrix are swapped, which results in. ${\displaystyle I_{n}}$ ${\displaystyle T_{1,2}}$

General properties

• Elementary row transformations (or column transformations) result from left multiplication (or right multiplication) with an elementary matrix.
• The rank of a matrix does not change by elementary row or column operations.
• If a linear system of equations is given in the form with and , then the following operations (made possible by multiplication with elementary matrices) do not change the solution and are therefore also called elementary transformations (where the operations on A and b are to be carried out simultaneously): ${\displaystyle (A,b)}$ ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle b\in K^{m\times 1}}$
  1. Adding the value of one row to another row. ${\displaystyle \alpha }$
  2. Swapping two lines.
  3. Multiplying a row by a non-zero value.

Group theoretical properties

Let it be the group of invertible n × n matrices . ${\displaystyle \mathrm {GL} _{n}(K)}$

• Elementary matrices are invertible, and so are the assignments

${\displaystyle K\to \mathrm {GL} _{n}(K),\quad \alpha \mapsto R_{ij}(\alpha )}$

such as

${\displaystyle K^{\times }\to \mathrm {GL} _{n}(K),\quad \alpha \mapsto S_{i}(\alpha )}$

are group homomorphisms . In particular,

${\displaystyle R_{ij}(\alpha )^{-1}=R_{ij}(-\alpha )}$

and

${\displaystyle S_{i}(\alpha )^{-1}=S_{i}(\alpha ^{-1}).}$

The matrices are their own inverses: ${\displaystyle T_{ij}}$

${\displaystyle T_{ij}^{-1}=T_{ij}.}$

• Every invertible matrix can be written as a product of elementary matrices, i. H. the elementary matrices generate the group . Type 1 and type 3 are sufficient for this. An important application of elementary matrices is based on this: In order to prove a statement for all invertible matrices, the following two points are sufficient: ${\displaystyle \mathrm {GL} _{n}(K)}$
  1. It applies to elementary matrices.
  2. If it applies to matrices A and B , it also applies to your product AB .

literature

• Gerd Fischer : Analytical Geometry (= Vieweg study. Vol. 35 Basic course in mathematics ). 4th revised edition. Vieweg, Braunschweig et al. 1985, ISBN 3-528-37235-4 , pp. 91-97.
• Gerd Fischer: Linear Algebra. An introduction for first-year students. 17th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 , pp. 163-173.

Web links

• Helmut Lenzing, Andrew Hubery, Markus Diekämper, Marc Jesse: Lineare Algebra I - winter semester 2003/2004 (lecture notes), Chapter III, Section 3.9 Elementary Matrices (pdf; 301 kB)
• Elementary Matrices on PlanetMath