Elementary matrix transformations
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Elementary matrix transformations or elementary row and column transformations are linear transformations which are normally used in Gaussian 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.
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1. Row-switching 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>
- That is, Tij is the matrix produced by exchanging row i and row j of the identity matrix.
Properties
- The matrix Tij is square.
- The inverse of this matrix is itself: Tij−1=Tij.
- Since the determinant of the identity matrix is unity, det[Tij] = −1. It follows that for any conformable square matrix A: det[TijA] = −det[A].
2. Row-multiplying transformations
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>
Properties
- The inverse of this matrix is: Ti(m)−1 = Ti(1/m).
- The matrix and its inverse are diagonal matrices.
- det[Ti(m)] = m. Therefore for a conformable square matrix A: det[Ti(m)A] = m det[A].
3. Linear combinator transformations
This transformation, Tij(m), subtracts 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>
Properties
- Tij(m)−1 = Tij(−m) (inverse matrix).
- The matrix and its inverse are triangular matrices.
- det[Tij(m)] = 1. Therefore, for a conformable square matrix A: det[Tij(m)A] = det[A].