In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an elementary matrix is a simple matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 which differs from the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

 by one single elementary row operation. The elementary matrices generate the general linear group

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

 of invertible matrices. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations.

In algebraic K-theory

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

, "elementary matrices" refers only to the row-addition matrices.

Use in solving systems of equations

Elementary row operations do not change the solution set of the system of linear equations represented by a matrix, and are used in Gaussian elimination

Gaussian elimination

In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...

 (respectively, Gauss-Jordan elimination) to reduce a matrix to row echelon form

Row echelon form

In linear algebra a matrix is in row echelon form if* All nonzero rows are above any rows of all zeroes, and...

 (respectively, reduced row echelon form).

The acronym "ERO" is commonly used for "elementary row operations".

Elementary row operations do not change the kernel of a matrix (and hence do not change the solution set), but they do change the
image

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

. Dually

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

, elementary column operations do not change the image, but they do change the kernel.

There are three types of (n x n) elementary matrices:
1) Permutation Matrix
2) Diagonal Matrix
3) Unipotent Matrix

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.

Row addition: A row can be replaced by the sum of that row and a multiple of another row.

The elementary matrix for any row operation is obtained by executing the operation on an identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

.

Row-switching transformations

This transformation, T_ij, switches all matrix elements on row i with their counterparts on row j. The matrix resulting in this transformation is obtained by swapping row i and row j of the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

.

That is, T_ij is the matrix produced by exchanging row i and row j of the identity matrix.

Properties

• The inverse of this matrix is itself: T_ij⁻¹=T_ij.
• Since the determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
   of the identity matrix is unity, det[T_ij] = −1. It follows that for any conformable square matrix A: det[T_ijA] = −det[A].

Row-multiplying transformations

This transformation, T_i(m), multiplies all elements on row i by m where m is non zero. The matrix resulting in this transformation is obtained by multiplying all elements of row i of the identity matrix by m.

Properties

• The inverse of this matrix is: T_i(m)⁻¹ = T_i(1/m).
• The matrix and its inverse are diagonal matrices
  Diagonal matrix
  In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
  
  .
• det[T_i(m)] = m. Therefore for a conformable square matrix A: det[T_i(m)A] = m det[A].

Row-addition transformations

This transformation, T_ij(m), adds row j multiplied by m to row i. The matrix resulting in this transformation is obtained by taking row j of the identity matrix, and adding it m times to row i.
These are also called shear mappings or transvections.

Properties

• T_ij(m)⁻¹ = T_ij(−m) (inverse matrix).
• The matrix and its inverse are triangular matrices
  Triangular matrix
  In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
  
  .
• det[T_ij(m)] = 1. Therefore, for a conformable
  Conformable matrix
  In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation .-Examples:...
  
   square matrix A: det[T_ij(1)A] = det[A].

Row-addition transforms satisfy the Steinberg relations.

See also

• Gaussian elimination
  Gaussian elimination
  In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
  
  
• Linear algebra
  Linear algebra
  Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
  
  
• System of linear equations
• Matrix (mathematics)
  Matrix (mathematics)
  In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
  
  
• LU decomposition
  LU decomposition
  In linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...