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Elementary matrix transformations

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:

  1. Row switching transformations,
  2. Row multiplying transformations,
  3. Linear combinator transformations.

Table of contents

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>

Properties

  • 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.

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).
  • When applied to a matrix A: det[TA]=mdet[A].
  • The matrix and it's inverse are lower triangular matrices.

3. Linear combinator transformations

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>

Properties

  • 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



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