Elementary Matrix

linear-algebra

Definition

Elementary Matrix

An elementary matrix is a square matrix that is obtained from applying single elementary column (or row) operations to the identity matrix.

Examples:

$$T^{\prime }=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& \lambda \\ 0& 0& 1\end{array}\right]$$

$T^{\prime }$ is an elementary matrix since an $\lambda$-multiple of the second column of the identity matrix $I_3$ is added to the third column.

$$T^{\prime \prime }=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 0& 1\end{array}\right]$$

$T^{\prime \prime }$ is an elementary matrix of since row one and row two are swapped.

Invertibility

Elementary matrices are invertible. For example:

$$\begin{array}{rl}(T^{\prime }{)}^{-1}& ={\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -\lambda \\ 0& 0& 1\end{array}\right]}^{-1}\\ & =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -\lambda \\ 0& 0& 1\end{array}\right]\end{array}$$

Thus, transforming a matrix $A$ into a matrix $A^{\prime }$ using single elementary column (or row) operations is equivalent to multiplication with an invertible matrix $T$:

$$A^{\prime }=A\cdot T$$