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Elementary matrix

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Matrix which differs from the identity matrix by one elementary row operation

Inmathematics, anelementary matrix is a squarematrix obtained from the application of a single elementary row operation to theidentity matrix. The elementary matrices generate thegeneral linear groupGL_n(F) whenF is afield. Left multiplication (pre-multiplication) by an elementary matrix represents the correspondingelementary row operation, while right multiplication (post-multiplication) represents the correspondingelementary column operation.

Elementary row operations are used inGaussian elimination to reduce a matrix torow echelon form. They are also used inGauss–Jordan elimination to further reduce the matrix toreduced row echelon form.

Elementary row operations

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There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

Row switching: A row within the matrix can be switched with another row.; $R_{i}\leftrightarrow R_{j}$

Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known asscaling a row.; $kR_{i}\rightarrow R_{i},\ {\mbox{where }}k\neq 0$

Row addition: A row can be replaced by the sum of that row and a multiple of another row.; $R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j$

IfE is an elementary matrix, as described below, to apply the elementary row operation to a matrixA, one multipliesA by the elementary matrix on the left,EA. The elementary matrix for any row operation is obtained by executing the operation on theidentity matrix. This fact can be understood as an instance of theYoneda lemma applied to the category of matrices.^[1]

Row-switching transformations

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Properties

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The inverse of this matrix is itself: $T_{i,j}^{-1}=T_{i,j}.$
Since thedeterminant of the identity matrix is unity, $\det(T_{i,j})=-1.$ It follows that for any square matrixA (of the correct size), we have $\det(T_{i,j}A)=-\det(A).$
For theoretical considerations, the row-switching transformation can be obtained from row-addition and row-multiplication transformations introduced below because $T_{i,j}=D_{i}(-1)\,L_{i,j}(-1)\,L_{j,i}(1)\,L_{i,j}(-1).$

Row-multiplying transformations

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The next type of row operation on a matrixA multiplies all elements on rowi bym wherem is a non-zeroscalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in theith position, where it ism.

D_{i}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&m&&&\\&&&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}

SoD_i(m)A is the matrix produced fromA by multiplying rowi bym.

Coefficient wise, theD_i(m) matrix is defined by :

[D_{i}(m)]_{k,l}={\begin{cases}0&k\neq l\\1&k=l,k\neq i\\m&k=l,k=i\end{cases}}

Properties

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The inverse of this matrix is given by $D_{i}(m)^{-1}=D_{i}\left({\tfrac {1}{m}}\right).$
The matrix and its inverse arediagonal matrices.
$\det(D_{i}(m))=m.$ Therefore, for a square matrixA (of the correct size), we have $\det(D_{i}(m)A)=m\det(A).$

Row-addition transformations

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The final type of row operation on a matrixA adds rowj multiplied by a scalarm to rowi. The corresponding elementary matrix is the identity matrix but with anm in the(i, j) position.

L_{ij}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&\ddots &&&\\&&m&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}

SoL_ij(m)A is the matrix produced fromA by addingm times rowj to rowi. AndA L_ij(m) is the matrix produced fromA by addingm times columni to columnj.

Coefficient wise, the matrixL_i,j(m) is defined by :

[L_{i,j}(m)]_{k,l}={\begin{cases}0&k\neq l,k\neq i,l\neq j\\1&k=l\\m&k=i,l=j\end{cases}}

Properties

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These transformations are a kind ofshear mapping, also known as atransvections.
The inverse of this matrix is given by $L_{ij}(m)^{-1}=L_{ij}(-m).$
The matrix and its inverse aretriangular matrices.
$\det(L_{ij}(m))=1.$ Therefore, for a square matrixA (of the correct size) we have $\det(L_{ij}(m)A)=\det(A).$
Row-addition transforms satisfy theSteinberg relations.

References

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^Perrone (2024), pp. 119–120

Axler, Sheldon Jay (1997),Linear Algebra Done Right (2nd ed.), Springer-Verlag,ISBN 0-387-98259-0
Lay, David C. (August 22, 2005),Linear Algebra and Its Applications (3rd ed.), Addison Wesley,ISBN 978-0-321-28713-7
Meyer, Carl D. (February 15, 2001),Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM),ISBN 978-0-89871-454-8, archived fromthe original on 2009-10-31
Perrone, Paolo (2024),Starting Category Theory, World Scientific,doi:10.1142/9789811286018_0005,ISBN 978-981-12-8600-1
Poole, David (2006),Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole,ISBN 0-534-99845-3
Anton, Howard (2005),Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
Leon, Steven J. (2006),Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
Strang, Gilbert (2016),Introduction to Linear Algebra (5th ed.), Wellesley-Cambridge Press,ISBN 978-09802327-7-6

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