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.
![In](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
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 .
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![n \ times p](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
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.
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![In](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![E _ {{i, j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7814421833a1b6fae2769bb795158a0c02ecda0)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![(i, j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
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.
![(i, j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712)
![\ alpha \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3581f6410248bed60440badc5bd362ca1cc8c82)
![i \ neq j](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
This is generated by
-
, where and is.![\ alpha \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3581f6410248bed60440badc5bd362ca1cc8c82)
![i \ neq j](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be)
For the abbreviation we write
![R _ {{i, j}} (\ alpha) = I_ {n} + \ alpha \ times E _ {{i, j}};](https://wikimedia.org/api/rest_v1/media/math/render/svg/897ad84c20f9874ffaa6482bf4f502d90ab48ffb)
please note, however, that this is not a standard notation.
So it is explicitly true
-
,
where is in the -th row and -th column.
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
Examples
![R _ {{2,1}} (- 7) = {\ begin {pmatrix} 1 & 0 & 0 \\ - 7 & 1 & 0 \\ 0 & 0 & 1 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd375a5699ded156d47bfa3ddf999348d22af95)
![R _ {{1,3}} (- 3) = {\ begin {pmatrix} 1 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5975aa1231a8a213637ad5453680acbb8a5be58)
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 .
![In](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
![i \ neq j](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be)
![In](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
![(i, i)](https://wikimedia.org/api/rest_v1/media/math/render/svg/899c08ccec9bf245d850a44ba8a5275bee6b3b8d)
![{\ displaystyle (j, j)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ca8093b6a5117dc3c2a0464aeb88004cce3c79)
![(i, j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712)
![(j, i)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f34fc97a634ea4ba7099ffc801a5268cc431bc6)
The following matrix operations do this:
-
, For
For abbreviation, we define type 2 here as
![T _ {{i, j}} = I_ {n} -E _ {{i, i}} - E _ {{j, j}} + E _ {{i, j}} + E _ {{j, i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c7f48b248e346a26db140442b99502d8f77559)
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 _ {(y, j)} & \\ &&&& \\\ end {pmatrix}} + {\ begin {pmatrix} &&&& \\ &&& 1 _ {(i, j)} & \\ &&&& \\ &&&& \\ &&&& \\\ end {pmatrix}} + {\ begin {pmatrix} &&&& \ \ &&&& \\ &&&& \\ & 1 _ {(j, i)} &&& \\ &&&& \\\ end {pmatrix}} =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f86a0ab0d8cbc88587eb72db2b853c9b4c3f3ff)
![= {\ begin {pmatrix} 1 &&&& \\ & 0 _ {{(i, i)}} & \ cdots & 1 _ {{(i, j)}} & \\ & \ vdots & 1 & \ vdots & \\ & 1 _ {{(j , i)}} & \ cdots & 0 _ {{(j, j)}} & \\ &&&& 1 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60def3fab28c0fc3f705bc3f398afd03ec4686ef)
The following example shows how the -th and -th lines are swapped:
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
example
![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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f79ec69cc288468363a237eceda95f2fc91ec4)
Is analog
![T _ {{2,4}} = {\ begin {pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ee2ba5ca61062b7b2ef3204fc8989e0cd2020a)
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.
![(i, i)](https://wikimedia.org/api/rest_v1/media/math/render/svg/899c08ccec9bf245d850a44ba8a5275bee6b3b8d)
![\ gamma \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/444b26e646c5c7e1030383377533b7c4655cd296)
This is achieved via
-
, with and![\ gamma \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/444b26e646c5c7e1030383377533b7c4655cd296)
(At this point , add 1 and subtract 1.)
![(i, i)](https://wikimedia.org/api/rest_v1/media/math/render/svg/899c08ccec9bf245d850a44ba8a5275bee6b3b8d)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
For abbreviation, type 3 should be used here as
![S_ {i} (\ gamma) = I_ {n} + (\ gamma -1) \ cdot E _ {{i, i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ef0b0301566bd0b5a2a6dd199f9bc093266893)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7515c0938332174918c97e3dc9f35137655d6a)
Examples
![S_ {2} (8) = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 1 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89cdbb9fe154159d78925460e505fe8ab3e73d32)
![S_ {3} (17) = {\ begin {pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 17 & 0 \\ 0 & 0 & 0 & 1 \\\ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cffeb2774558c84b78672a2121b6c47cbda72a)
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.
![m \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
![R _ {{i, j}} (\ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d609f8a3476510e3a23846aed9dc16baaccb70db)
![T _ {{i, j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8cf80f0bd7e84249d866cab104dfc625bb7fca)
![S_ {i} (\ gamma)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8740bca82558fd768d74f7929b2d744cb5318822)
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)![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
-
adds the -fold of the j-th row of A to the i-th row of A. (EZU II)![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
-
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)![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
-
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.![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
-
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.
![In](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
![T _ {{1,2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6b6fc8ee29ae2d77c5de1b8cc164167e5d7a4b)
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):
![(From)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5730a05f5d3a0b15d2e6ccb3b2c2e31fd79cbb)
![A \ in K ^ {m \ times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab303433d6aa889f1f94258f6afb6a04ffea388d)
![b \ in K ^ {{m \ times 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/336bf495975e7041c1effa615c712b8752d300cb)
- Adding the value of one row to another row.
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
- Swapping two lines.
- Multiplying a row by a non-zero value.
Group theoretical properties
Let it be the group of invertible n × n matrices .
![{\ mathrm {GL}} _ {n} (K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b927823796448834198a1d33959863292d91e6b4)
- Elementary matrices are invertible, and so are the assignments
![K \ to {\ mathrm {GL}} _ {n} (K), \ quad \ alpha \ mapsto R _ {{ij}} (\ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7e29149543bbfa84a5953e202a4f4fb92cbe1e)
- such as
![K ^ {\ times} \ to {\ mathrm {GL}} _ {n} (K), \ quad \ alpha \ mapsto S_ {i} (\ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/74cd213db61b48983049422010666e50eafb3609)
- are group homomorphisms . In particular,
![R _ {{ij}} (\ alpha) ^ {{- 1}} = R _ {{ij}} (- \ alpha)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1734673c33f624c6554e591026440945f1a2a70)
- and
![S_ {i} (\ alpha) ^ {{- 1}} = S_ {i} (\ alpha ^ {{- 1}}).](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd4ff866840b88713aaca1f46c0b52a77af5ee0)
- The matrices are their own inverses:
![T_ {ij}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9de5760ded748359e36c7fb067c45f5e9642e890)
![T _ {{ij}} ^ {{- 1}} = T _ {{ij}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f399a93cec27b676eb16552ef9e98286cd4015)
- 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:
![{\ mathrm {GL}} _ {n} (K)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b927823796448834198a1d33959863292d91e6b4)
- 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