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.

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:

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

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.
• When applied to a matrix A: det[T_ijA]=-det[A].
• The matrix T_ij is square.

2. Row multiplying transformations

This transformation, T_i(m), multiplies all elements on row i with m. The matrix resulting in this transformation is:

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

Properties

• The inverse of this matrix is: T_i(m)^-1=T_i(1/m).
• When applied to a matrix A: det[TA]=mdet[A].
• The matrix and its inverse are lower triangular matrices.

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:

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

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 its inverse are lower triangular matrices.

See also

• Linear algebra
• Gauss elimination method