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In mathematics, an elementary matrix is a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL''n''(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to
row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian ...
. They are also used in
Gauss–Jordan elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
to further reduce the matrix to
reduced row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian e ...
.


Elementary row operations

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 as ''scaling'' a row. : kR_i \rightarrow R_i,\ \mbox 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 i \neq j If ''E'' is an elementary matrix, as described below, to apply the elementary row operation to a matrix ''A'', one multiplies ''A'' by the elementary matrix on the left, ''EA''. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix. This fact can be understood as an instance of the
Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...
applied to the category of matrices.


Row-switching transformations

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. :T_ = \begin 1 & & & & & & \\ & \ddots & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & \\ & & 1 & & 0 & & \\ & & & & & \ddots & \\ & & & & & & 1 \end So ''T''''ij''''A'' is the matrix produced by exchanging row ''i'' and row ''j'' of ''A''. Coefficient wise, the matrix T_ is defined by : : _ = \begin 0 & k \neq i, k \neq j ,k \neq l \\ 1 & k \neq i, k \neq j, k = l\\ 0 & k = i, l \neq j\\ 1 & k = i, l = j\\ 0 & k = j, l \neq i\\ 1 & k = j, l = i\\ \end


Properties

* The inverse of this matrix is itself: ''T''''ij''−1 = ''T''''ij''. * Since the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the identity matrix is unity, det(''T''''ij'') = −1. It follows that for any square matrix ''A'' (of the correct size), we have det(''T''''ij''''A'') = −det(''A'').


Row-multiplying transformations

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 Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
(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''. :D_i(m) = \begin 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & m & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1 \end So ''D''''i''(''m'')''A'' is the matrix produced from ''A'' by multiplying row ''i'' by ''m''. Coefficient wise, the D_i(m) matrix is defined by : :
_i(m) I, or i, is the ninth Letter (alphabet), letter and the third vowel letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in Engl ...
= \begin 0 & k \neq l \\ 1 & k = l, k \neq i \\ m & k = l, k= i \end


Properties

* The inverse of this matrix is given by ''D''''i''(''m'')−1 = ''D''''i''(1/''m''). * The matrix and its inverse are
diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
. * det(''D''''i''(''m'')) = ''m''. Therefore for a square matrix ''A'' (of the correct size), we have det(''D''''i''(''m'')''A'') = ''m'' det(''A'').


Row-addition transformations

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. :L_(m) = \begin 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & \ddots & & & \\ & & m & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1 \end So ''L''''ij''(''m'')''A'' is the matrix produced from ''A'' by adding ''m'' times row ''j'' to row ''i''. And ''A'' ''L''''ij''(''m'') is the matrix produced from ''A'' by adding ''m'' times column ''i'' to column ''j''. Coefficient wise, the matrix L_(m) is defined by : : _(m) = \begin 0 & k \neq l, k \neq i, l \neq j \\ 1 & k = l \\ m & k = i, l = j \end


Properties

* These transformations are a kind of
shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
, also known as a ''transvections''. * The inverse of this matrix is given by ''L''''ij''(''m'')−1 = ''L''''ij''(−''m''). * The matrix and its inverse are triangular matrices. * det(''L''''ij''(''m'')) = 1. Therefore, for a square matrix ''A'' (of the correct size) we have det(''L''''ij''(''m'')''A'') = det(''A''). * Row-addition transforms satisfy the Steinberg relations.


See also

* Gaussian elimination *
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
* System of linear equations *
Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...
*
LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a p ...
*
Frobenius matrix A Frobenius matrix is a special kind of square matrix from numerical mathematics. A matrix is a Frobenius matrix if it has the following three properties: * all entries on the main diagonal are ones * the entries below the main diagonal of at most ...


References

* * * * * * * {{DEFAULTSORT:Elementary Matrix Linear algebra