Elementary matrix

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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.

Operations[edit]

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, to 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[edit]

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

Properties[edit]

  • The inverse of this matrix is itself: Tij−1=Tij.
  • Since the determinant of the identity matrix is unity, det[Tij] = −1. It follows that for any square matrix A (of the correct size), we have det[TijA] = −det[A].

Row-multiplying transformations[edit]

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 Ti(m)⋅A is the matrix produced from A by multiplying row i by m.

Properties[edit]

  • The inverse of this matrix is: Ti(m)−1 = Ti(1/m).
  • The matrix and its inverse are diagonal matrices.
  • det[Ti(m)] = m. Therefore for a square matrix A (of the correct size), we have det[Ti(m)A] = m det[A].

Row-addition transformations[edit]

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

Properties[edit]

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

See also[edit]

References[edit]

  • 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