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 1. Row switching transformations 1.1 Properties 2 2. Row multiplying transformations 2.1 Properties 3 3. Linear combinator transformations 3.1 Properties

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:

<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: 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.

2. Row multiplying transformations

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>

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 it's inverse are lower triangular matrices.

3. Linear combinator transformations

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>

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 it's inverse are lower triangular matrices.

See also

• Linear algebra
• Gauss elimination method