Elementary matrix transformations: Difference between revisions

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#REDIRECT [[Elementary matrix]]

[[ja:行列の基本変形]]

[[de:Elementarmatrix]]

'''Elementary matrix transformations''' or '''Elementary 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.

[[is:Frumfylki]]

[[ja:行列の基本変形]]

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

[[zh:初等變換]]

# '''Row switching''' transformations,

# '''Row multiplying''' transformations,

# '''Linear combinator''' 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>

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^-1=T_ij''.

:*When applied to a matrix ''A'': ''det[TA]=-det[A]''.

:*The matrix T is square.

===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>

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)^-1=T_i(1/m)''.

:*When applied to a matrix ''A'': ''det[TA]=mdet[A]''.

:*The matrix and its inverse are lower triangular matrices.

===3. Linear combinator transformations===

This transformation, ''T_ij(m)'', subtracts row i multiplied by ''m'' from row j. The matrix resulting in this transformation is:<br>

:<math>

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

</math>

====Properties====

:*The inverse of this matrix is: ''T_ij(m)^-1=T_ij(-m)''.

:*When applied to a matrix ''A'': ''det[TA]=det[A]''.

:*The matrix and its inverse are lower triangular matrices.

'''See also'''

*[[Linear algebra]]

*[[Gauss elimination method]]