Elementary matrix transformations or
Elementary row and column transformations are
linear transformations which are normally used in
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.
This transformation,
Tij, switches all matrix elements on row i with their counterparts on row j. The matrix resulting in this transformation is:
- <math>
T_{i,j} = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>
- The inverse of this matrix is itself: Tij-1=Tij.
- When applied to a matrix A: det[TA]=-det[A].
- The matrix and it's inverse are lower triangular matrices.
This transformation,
Ti(m), multiplies all elements on row i with
m. The matrix resulting in this transformation is:
- <math>
T_i(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & m & & & & \\ & & & & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>
- The inverse of this matrix is: Ti(m)-1=Ti(1/m).
- When applied to a matrix A: det[TA]=mdet[A].
- The matrix and it's inverse are lower triangular matrices.
This transformation,
Tij(m), substracts row i multiplied by
m from row j. The matrix resulting in this transformation is:
- <math>
T_{i,j}(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & \ddots & & & & \\ & & -m & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}
</math>
- The inverse of this matrix is: Tij(m)-1=Tij(-m).
- When applied to a matrix A: det[TA]=det[A].
- The matrix and it's inverse are lower triangular matrices.
See also