Elementary matrix transformations: Difference between revisions

Revision as of 18:35, 12 October 2004

Revision as of 03:08, 3 December 2004

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[[ja:行列の基本変形]]

'''Elementary matrix transformations''' or '''Elementaryelementary row and column transformations''' are [[Linear transformation|linear transformations]] which are normally used in [[Gauss elimination method|gauss elimination]] to solve a set of linear equations.

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

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

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

# '''Linear combinator''' transformations.

===1. Row -switching transformations===

===1. 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:<br>

:<math>

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:*Since the [[Determinant|determinant]] of the [[Identity_matrix|identity matrix]] is unity, ''det[T_ij]=-1''. It follows that for any conformable square matrix ''A'': ''det[T_ijA]=-det[A]''.

===2. Row -multiplying transformations===

This transformation, ''T_i(m)'', multiplies all elements on row i with ''m''. The matrix resulting in this transformation is:<br>

:<math>

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:*''det[T_ij(m)]=1''. Therefore, for a conformable square matrix ''A'': ''det[T_ij(m)A]=det[A]''.

'''==See also'''==

'''See also'''

*[[Linear algebra]]

*[[Gauss elimination method]]