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.
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 .
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.
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.
This is generated by
-
, where and is.
For the abbreviation we write
please note, however, that this is not a standard notation.
So it is explicitly true
-
,
where is in the -th row and -th column.
Examples
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 .
The following matrix operations do this:
-
, For
For abbreviation, we define type 2 here as
The operations generally look like this:
The following example shows how the -th and -th lines are swapped:
example
Is analog
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.
This is achieved via
-
, with and
(At this point , add 1 and subtract 1.)
For abbreviation, type 3 should be used here as
To be defined. Again, it is not a standard notation.
Operations performed:
Examples
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.
Multiplication from the left gives line transformations:
-
multiplies the i-th line of A by the value , whereby the remaining lines remain unchanged (EZU I)
-
adds the -fold of the j-th row of A to the i-th row of A. (EZU II)
-
swaps the i-th row of A with the j-th row of A. (EZU III)
Multiplication from the right gives column transformations:
-
multiplies the i-th column of A by the value , leaving the remaining columns unchanged. (ESU I)
-
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.
-
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.
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):
- Adding the value of one row to another row.
- Swapping two lines.
- Multiplying a row by a non-zero value.
Group theoretical properties
Let it be the group of invertible n × n matrices .
- Elementary matrices are invertible, and so are the assignments
- such as
- are group homomorphisms . In particular,
- and
- The matrices are their own inverses:
- 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:
- It applies to elementary matrices.
- 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