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,
T_{ij}, 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: T_{ij}^{-1}=T_{ij}.
- When applied to a matrix A: det[TA]=-det[A].
- The matrix and it's inverse are lower triangular matrices.
This transformation,
T_{i}(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: T_{i}(m)^{-1}=T_{i}(1/m).
- When applied to a matrix A: det[TA]=mdet[A].
- The matrix and it's inverse are lower triangular matrices.
This transformation,
T_{ij}(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: T_{ij}(m)^{-1}=T_{ij}(-m).
- When applied to a matrix A: det[TA]=det[A].
- The matrix and it's inverse are lower triangular matrices.
See also
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