Elementary matrix transformations: Difference between revisions

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{{mergewith|#REDIRECT [[Elementary operations|date=Julyrow 2006}}operations]]

'''Elementary matrix transformations''' or '''elementary row and column transformations''' are [[linear transformation]]s which are normally used in [[Gaussian elimination]] to solve a set of [[linear equation]]s. When a matrix is multiplied by an elementary matrix, the matrix would be transformed the same as if we did the operation itself. For example, when we multiply a matrix by the row-switching transformation elementary matrix, the matrix would switch the rows much like if you did the row-switching manually.

We distinguish three types of elementary transformations and their corresponding matrices:

# '''Row-switching''' transformations,

# '''Row-multiplying''' transformations,

# '''Linear combinator''' transformations.

Column transformations may be defined similarly.

==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]].

:<math>

T_{i,j} = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>

: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^&minus;1=T_ij''.

:*Since the [[determinant]] of the identity matrix is unity, det[''T''_''ij''] = &minus;1. It follows that for any [[conformable]] square matrix ''A'': det[''T''_''ij''''A''] = &minus;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''.

:<math>

T_i(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & m & & & & \\ & & & & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>

===Properties===

:*The inverse of this matrix is: ''T_i''(''m'')^&minus;1 = ''T_i''(1/''m'').

:*The matrix and its inverse are [[diagonal matrix|diagonal matrices]].

:*det[''T''_''i''(m)] = ''m''. Therefore for a conformable square matrix ''A'': det[''T''_''i''(''m'')''A''] = ''m'' det[''A''].

==Linear combinator transformations==

This transformation, ''T_ij''(''m''), subtracts row ''j'' multiplied by ''m'' from row ''i''. The matrix resulting in this transformation is obtained by taking row ''j'' of the identity matrix, and subtracting from it ''m'' times row ''i''.

:<math>

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

</math>

===Properties===

:*''T_ij''(''m'')^&minus;1 = ''T_ij''(&minus;''m'') (inverse matrix).

:*The matrix and its inverse are [[triangular matrix|triangular matrices]].

:*det[''T_ij''(''m'')] = 1. Therefore, for a [[conformable matrix|conformable]] square matrix ''A'': det[''T''_''ij''(''m'')''A''] = det[''A''].

==Use of these transformations==

These transformations, the matrices of which are called elementary matrices, are the ones which convert a given system of equations, into one suitable for obtaining the solution in the Gauss elimination method. Directly applying any of these transforms on a matrix is equivalent to pre multiplying (post multiplying) that matrix by the corresponding row transformation (column transformation).

==See also==

*[[Linear algebra]]

*[[Gauss elimination method]]

[[Category:Matrix theory]]

[[de:Elementarmatrix]]