In
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.
...
, 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 ...
is in echelon form if it has the shape resulting from a
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 ...
.
A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and
column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.
Specifically, a matrix is in row echelon form if
* All rows consisting of only zeroes are at the bottom.
* The
leading entry
In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
(that is the left-most nonzero entry) of every nonzero row is to the right the leading entry of every row above.
Some texts add the condition that the leading coefficient must be 1 while others regard this as ''reduced'' row echelon form.
These two conditions imply that all entries in a column below a leading coefficient are zeros.
The following is an example of a 4x5 matrix in row echelon form, which is not in ''reduced'' row echelon form (see below):
:
Many properties of matrices may be easily deduced from their row echelon form, such as the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
and the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
.
Reduced row echelon form
A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
* It is in row echelon form.
* The leading entry in each nonzero row is a 1 (called a leading 1).
* Each column containing a leading 1 has zeros in all its other entries.
The reduced row echelon form of a matrix may be computed by
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 ...
. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it.
For a given matrix, despite the row echelon form not being unique, all row echelon forms and the reduced row echelon form have the same number of zero rows and the pivots are located in the same indices.
This is an example of a matrix in reduced row echelon form, which shows that the left part of the matrix is not always an
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 ...
:
: