Elementary matrix

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of 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".

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.

In algebraic K-theory, "elementary matrices" refers only to the row-addition matrices.

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.

$kR_{i}\rightarrow R_{i},\ {\mbox{where }}k\neq 0$

Row addition

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

$R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j$

If E is an elementary matrix, as described below, two 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.

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.

$T_{i,j}={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&0&&1&&\\&&&\ddots &&&&\\&&1&&0&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}}\quad$

So T_ij⋅A is the matrix produced by exchanging row i and row j of A.

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_ijA] = −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 ith position, where it is m.

$T_{i}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&m&&&\\&&&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}\quad$

So T_i(m)⋅A is the matrix produced from A by multiplying row i 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.
• det[T_i(m)] = m. Therefore for a square matrix A (of the correct size), we have det[T_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.

$T_{i,j}(m)={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&1&&&&&\\&&&\ddots &&&&\\&&m&&1&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}}$

So T_i,j(m)⋅A is the matrix produced from A by adding m times row j to row i.

Properties

• These transformation are a kind of shear mapping, also known as a transvections.
• T_ij(m)⁻¹ = T_ij(−m) (inverse matrix).
• The matrix and its inverse are triangular matrices.
• det[T_ij(m)] = 1. Therefore, for a square matrix A (of the correct size) we have det[T_ij(m)A] = det[A].
• Row-addition transforms satisfy the Steinberg relations.

See also

• Gaussian elimination
• Linear algebra
• System of linear equations
• Matrix (mathematics)
• LU decomposition

References

See also: Linear algebra § Further reading

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0
• Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7
• Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8
• Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall