Elementary matrix transformations

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. When a matrix is multiplied by an elementary matrix, the matrix would be transformed the same as if we did the operation itself. For example, when we multiply a matrix by the row-switching transformation elementary matrix, the matrix would switch the rows much like if you did the row-switching manually.

We distinguish three types of elementary transformations and their corresponding matrices:

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

Column transformations may be defined similarly.

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 obtained by swapping row i and row j of the identity matrix.

${\displaystyle T_{i,j}={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&0&&1&&\\&&&\ddots &&&&\\&&1&&0&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}},\quad }$ 

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

Properties

• 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].

Row-multiplying transformations

This transformation, T_i(m), multiplies all elements on row i by m where m is non zero. The matrix resulting in this transformation is obtained by multiplying all elements of row i of the identity matrix by m.

${\displaystyle T_{i}(m)={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&1&&&&&\\&&&m&&&&\\&&&&&1&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}},\quad }$ 

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].

Linear combinator transformations

This transformation, T_ij(m), subtracts row j multiplied by m from row i. The matrix resulting in this transformation is obtained by taking row j of the identity matrix, and subtracting from it m times row i.

${\displaystyle T_{i,j}(m)={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&1&&&&&\\&&&\ddots &&&&\\&&-m&&1&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}}}$ 

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].

Use of these transformations

These transformations, the matrices of which are called elementary matrices, are the ones which convert a given system of equations, into one suitable for obtaining the solution in the Gauss elimination method. Directly applying any of these transforms on a matrix is equivalent to pre multiplying (post multiplying) that matrix by the corresponding row transformation (column transformation).

See also

• Linear algebra
• Gauss elimination method