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| In [[mathematics]], an '''elementary matrix''' is a [[Matrix (mathematics)|matrix]] which differs from the [[identity matrix]] by one single elementary row operation. The elementary matrices generate the [[general linear group]] of [[invertible matrix|invertible matrices]]. Left multiplication (pre-multiplication) by an elementary matrix represents '''elementary row operations''', while right multiplication (post-multiplication) represents '''elementary column operations'''. The acronym "ERO" is commonly used for "elementary row operations".
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| Elementary row operations are used in [[Gaussian elimination]] to reduce a matrix to [[row echelon form]]. They are also used in [[Gauss-Jordan elimination]] to further reduce the matrix to [[reduced row echelon form]].
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| ==Operations==
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| There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):
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| ;Row switching: A row within the matrix can be switched with another row.
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| : <math>R_i \leftrightarrow R_j</math>
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| ;Row multiplication: Each element in a row can be multiplied by a non-zero constant.
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| : <math>kR_i \rightarrow R_i,\ \mbox{where } k \neq 0</math>
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| ;Row addition: A row can be replaced by the sum of that row and a multiple of another row.
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| : <math>R_i + kR_j \rightarrow R_i, \mbox{where } i \neq j </math>
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| If ''E'' is an elementary matrix, as described below, to apply the elementary row operation to a matrix ''A'', one multiplies the elementary matrix on the left, ''E⋅A''. The elementary matrix for any row operation is obtained by executing the operation on the [[identity matrix]].
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| ===Row-switching transformations===
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| The first type of row operation on a matrix ''A'' switches all matrix elements on row ''i'' with their counterparts on row ''j''. The corresponding elementary matrix is obtained by swapping row ''i'' and row ''j'' of the [[identity matrix]].
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| :<math>
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| T_{i,j} = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}\quad </math>
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| :So ''T<sub>ij</sub>⋅A'' is the matrix produced by exchanging row ''i'' and row ''j'' of ''A''.
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| ====Properties====
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| :*The inverse of this matrix is itself: ''T<sub>ij</sub><sup>−1</sup>=T<sub>ij</sub>''.
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| :*Since the [[determinant]] of the identity matrix is unity, det[''T''<sub>''ij''</sub>] = −1. It follows that for any square matrix ''A'' (of the correct size), we have det[''T''<sub>''ij''</sub>''A''] = −det[''A''].
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| ===Row-multiplying transformations===
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| The next type of row operation on a matrix ''A'' multiplies all elements on row ''i'' by ''m'' where ''m'' is a non-zero [[scalar (mathematics)|scalar]] (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ''i''th position, where it is ''m''.
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| :<math>
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| T_i(m) = \begin{bmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & m & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1\end{bmatrix}\quad </math>
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| :So ''T<sub>i</sub>(m)⋅A'' is the matrix produced from ''A'' by multiplying row ''i'' by ''m''.
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| ====Properties====
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| :*The inverse of this matrix is: ''T<sub>i</sub>''(''m'')<sup>−1</sup> = ''T<sub>i</sub>''(1/''m'').
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| :*The matrix and its inverse are [[diagonal matrix|diagonal matrices]].
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| :*det[''T''<sub>''i''</sub>(m)] = ''m''. Therefore for a square matrix ''A'' (of the correct size), we have det[''T''<sub>''i''</sub>(''m'')''A''] = ''m'' det[''A''].
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| ===Row-addition transformations===
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| The final type of row operation on a matrix ''A'' adds row ''j'' multiplied by a scalar ''m'' to row ''i''. The corresponding elementary matrix is the identity matrix but with an ''m'' in the (''i,j'') position.
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| :<math> | |
| T_{i,j}(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & \ddots & & & & \\ & & m & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}
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| </math>
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| :So ''T<sub>i,j</sub>(m)⋅A'' is the matrix produced from ''A'' by adding ''m'' times row ''j'' to row ''i''.
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| ====Properties====
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| :*These transformations are a kind of [[shear mapping]], also known as a ''transvections''.
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| :*''T<sub>ij</sub>''(''m'')<sup>−1</sup> = ''T<sub>ij</sub>''(−''m'') (inverse matrix).
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| :*The matrix and its inverse are [[triangular matrix|triangular matrices]].
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| :*det[''T<sub>ij</sub>''(''m'')] = 1. Therefore, for a square matrix ''A'' (of the correct size) we have det[''T''<sub>''ij''</sub>(''m'')''A''] = det[''A''].
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| :*Row-addition transforms satisfy the [[Steinberg relations]].
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| ==See also==
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| *[[Gaussian elimination]]
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| *[[Linear algebra]]
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| *[[System of linear equations]]
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| *[[Matrix (mathematics)]]
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| *[[LU decomposition]]
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| *[[Frobenius matrix]]
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| ==References==
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| {{See also|Linear algebra#Further reading}}
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| * {{Citation
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| | last = Axler
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| | first = Sheldon Jay
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| | date = 1997
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| | title = Linear Algebra Done Right
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| | publisher = Springer-Verlag
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| | edition = 2nd
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| | isbn = 0-387-98259-0
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| }}
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| * {{Citation
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| | last = Lay
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| | first = David C.
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| | date = August 22, 2005
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| | title = Linear Algebra and Its Applications
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| | publisher = Addison Wesley
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| | edition = 3rd
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| | isbn = 978-0-321-28713-7
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| }}
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| * {{Citation
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| | last = Meyer
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| | first = Carl D.
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| | date = February 15, 2001
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| | title = Matrix Analysis and Applied Linear Algebra
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| | publisher = Society for Industrial and Applied Mathematics (SIAM)
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| | isbn = 978-0-89871-454-8
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| | url = http://www.matrixanalysis.com/DownloadChapters.html
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| }}
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| * {{Citation
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| | last = Poole
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| | first = David
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| | date = 2006
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| | title = Linear Algebra: A Modern Introduction
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| | publisher = Brooks/Cole
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| | edition = 2nd
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| | isbn = 0-534-99845-3
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| }}
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| * {{Citation
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| | last = Anton
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| | first = Howard
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| | date = 2005
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| | title = Elementary Linear Algebra (Applications Version)
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| | publisher = Wiley International
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| | edition = 9th
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| }}
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| * {{Citation
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| | last = Leon
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| | first = Steven J.
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| | date = 2006
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| | title = Linear Algebra With Applications
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| | publisher = Pearson Prentice Hall
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| | edition = 7th
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| }}
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| {{DEFAULTSORT:Elementary Matrix}}
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| [[Category:Linear algebra]]
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Hello and welcome. My title is Irwin and I totally dig that title. For a whilst I've been in South Dakota and my parents live nearby. My working day occupation is a librarian. To gather coins is what her family members and her appreciate.
Review my blog home std test