Elementary matrix

In mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices generate the general linear group Template:Math when Template:Math 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 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.

$R_i\leftrightarrow R_j$

Row multiplication

Each element in a row can be multiplied by a non-zero constant. It is also known as scaling a row.

$k{R}_i\to R_i,\text{where }k\ne 0$

Row addition

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

$R_i+k{R}_j\to R_i,\text{where }i\ne j$

If Template:Mvar is an elementary matrix, as described below, to apply the elementary row operation to a matrix Template:Mvar, one multiplies Template:Mvar by the elementary matrix on the left, Template:Mvar. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix. This fact can be understood as an instance of the Yoneda lemma applied to the category of matrices.

Row-switching transformations

Template:See also The first type of row operation on a matrix Template:Mvar switches all matrix elements on row Template:Mvar with their counterparts on a different row Template:Mvar. The corresponding elementary matrix is obtained by swapping row Template:Mvar and row Template:Mvar of the identity matrix.

$T_{i,j}=\left[\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & & \\ & & 0& & 1& & \\ & & & \ddots & & & \\ & & 1& & 0& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right]$

So Template:Mvar is the matrix produced by exchanging row Template:Mvar and row Template:Mvar of Template:Mvar.

Coefficient wise, the matrix Template:Mvar is defined by :

$[T_{i,j}{]}_{k,l}=\{\begin{array}{cc}0& k\ne i,k\ne j,k\ne l\\ 1& k\ne i,k\ne j,k=l\\ 0& k=i,l\ne j\\ 1& k=i,l=j\\ 0& k=j,l\ne i\\ 1& k=j,l=i\end{array}$

Properties

• The inverse of this matrix is itself: $T_{i,j}^{-1}=T_{i,j}.$
• Since the determinant of the identity matrix is unity, $\det (T_{i,j})=-1.$ It follows that for any square matrix Template:Mvar (of the correct size), we have $\det (T_{i,j}A)=-\det (A).$
• For theoretical considerations, the row-switching transformation can be obtained from row-addition and row-multiplication transformations introduced below because $T_{i,j}=D_i(-1)L_{i,j}(-1)L_{j,i}(1)L_{i,j}(-1).$

Row-multiplying transformations

The next type of row operation on a matrix Template:Mvar multiplies all elements on row Template:Mvar by Template:Mvar where Template:Mvar 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 Template:Mvarth position, where it is Template:Mvar.

$D_i(m)=\left[\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & m& & & \\ & & & & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right]$

So Template:Math is the matrix produced from Template:Mvar by multiplying row Template:Mvar by Template:Mvar.

Coefficient wise, the Template:Math matrix is defined by :

$[D_i(m){]}_{k,l}=\{\begin{array}{cc}0& k\ne l\\ 1& k=l,k\ne i\\ m& k=l,k=i\end{array}$

Properties

• The inverse of this matrix is given by $D_i(m{)}^{-1}=D_i\left(\frac{1}{m}\right).$
• The matrix and its inverse are diagonal matrices.
• $\det (D_i(m))=m.$ Therefore, for a square matrix Template:Mvar (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 Template:Mvar adds row Template:Mvar multiplied by a scalar Template:Mvar to row Template:Mvar. The corresponding elementary matrix is the identity matrix but with an Template:Mvar in the Template:Math position.

$L_{ij}(m)=\left[\begin{array}{ccccccc}1& & & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & \ddots & & & \\ & & m& & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right]$

So Template:Math is the matrix produced from Template:Mvar by adding Template:Mvar times row Template:Mvar to row Template:Mvar. And Template:Math is the matrix produced from Template:Mvar by adding Template:Mvar times column Template:Mvar to column Template:Mvar.

Coefficient wise, the matrix Template:Math is defined by :

$[L_{i,j}(m){]}_{k,l}=\{\begin{array}{cc}0& k\ne l,k\ne i,l\ne j\\ 1& k=l\\ m& k=i,l=j\end{array}$

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.
• $\det (L_{ij}(m))=1.$ Therefore, for a square matrix Template:Mvar (of the correct size) we have $\det (L_{ij}(m)A)=\det (A).$
• Row-addition transforms satisfy the Steinberg relations.

See also

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

References

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