Elementary matrices

Matrices in the form $$\mathbf{I-uv}^T$$ where $$\mathbf{u, v}$$ are $$n\times 1$$ columns such that $$v^Tu \neq 1$$ are called elementary matrices (page 131 ). Elementary matrices are nonsingular, and using the Woodbury formula, it can be shown that:

$$\left(\mathbf{I-uv}^T\right)^{-1} = \mathbf{I}-\frac{\mathbf{uv}^T}{\mathbf{v}^T\mathbf u-1}$$

Three types of elementary matrices are of particular interest:
 * Type I: Interchanging rows (columns) $$i$$ and $$j$$
 * Type II: Multiplying row (column) $$i$$ by $$\alpha\neq 0$$
 * Type III is adding a multiple of row (column) $$i$$ to row (column) $$j$$

When applied from left, an elementary matrix of Type I, II or III does row operations (LR) and when applied from right, it applies column operations (RC)

Examples
The following are three elementary matrices corresponding to the three types listed above:

$$\mathbf E_{1}=\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$, $$\mathbf E_{2}=\begin{bmatrix}1 & 0 & 0 \\ 0 & \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$$, $$\mathbf E_{3}=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ \alpha & 0 & 1\end{bmatrix}$$

These three matrices can be written in to the canonical form as $$\mathbf E_1 = \mathbf{I-uu}^t$$ where $$\mathbf u=\mathbf e_1 - \mathbf e_2$$, $$\mathbf E_2 = \mathbf I-(1-\alpha)\mathbf e_2 \mathbf e_2^T$$, and $$\mathbf E_3 = \mathbf I \alpha + \mathbf e_3 \mathbf e_1^T$$