In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
by one single elementary row operation. The elementary matrices generate the
general linear group 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
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 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.
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.
:
;Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known as ''scaling'' a row.
:
;Row addition: A row can be replaced by the sum of that row and a multiple of another row.
:
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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
. This fact can be understood as an instance of the
Yoneda lemma 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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
.
:
So ''T''
''ij''''A'' is the matrix produced by exchanging row ''i'' and row ''j'' of ''A''.
Coefficient wise, the matrix
is defined by :
:
Properties
* The inverse of this matrix is itself: ''T''
''ij''−1 = ''T''
''ij''.
* Since the
determinant 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 (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''.
:
So ''D''
''i''(''m'')''A'' is the matrix produced from ''A'' by multiplying row ''i'' by ''m''.
Coefficient wise, the
matrix is defined by :
:
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.
* 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.
:
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
is defined by :
:
Properties
* These transformations are a kind of
shear mapping, 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
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
.
* 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 In algebraic K-theory, a field of mathematics, the Steinberg group \operatorname(A) of a ring A is the universal central extension of the commutator subgroup of the stable general linear group of A .
It is named after Robert Steinberg, and it ...
.
See also
*
Gaussian 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 ...
*
Linear algebra
*
System of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
*
Matrix (mathematics)
*
LU decomposition
*
Frobenius matrix
References
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{{DEFAULTSORT:Elementary Matrix
Linear algebra