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:

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

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 4 See also

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>

That is, T_ij is the matrix produced by exchanging row i and row j of the identity matrix.

Properties

• The matrix T_ij is square.
• The inverse of this matrix is itself: T_ij⁻¹=T_ij.
• Since the determinant of the identity matrix is unity, det[T_ij] = −1. It follows that for any conformable square matrix A: det[T_ijA] = −det[A].

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).
• The matrix and its inverse are diagonal matrices.
• det[T_i(m)] = m. Therefore for a conformable square matrix A: det[T_i(m)A] = m det[A].

3. Linear combinator transformations

This transformation, T_ij(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

• T_ij(m)⁻¹ = T_ij(−m) (inverse matrix).
• The matrix and its inverse are triangular matrices.
• det[T_ij(m)] = 1. Therefore, for a conformable square matrix A: det[T_ij(m)A] = det[A].

See also

• Linear algebra
• Gauss elimination methodde:Elementarmatrix

is:Frumfylki ja:行列の基本変形