A Little Summary

Let us summarise what we have learnt about Jacobi and Householder rotations.

Jacobi rotations are designed to annihilate a selected off-diagonal element, say, a_pq. Given matrix elements a_qq, a_pp, and a_pq the rotation itself can be calculated as follows:

$$\begin{eqnarray*}\theta &=& \cot{2\varphi} = \frac{a_{qq} - a_{pp}}{2a_{pq}} \\ ... ...ght) \\ v'_{rq} &=& v_{rq} + s\left(v_{rp} - \tau v_{rq}\right) \end{eqnarray*}$$

Householder rotations are designed to rotate the whole under diagonal column or the whole right-to-diagonal row onto its first direction. Given a subdiagonal column:

$$\boldsymbol{x} = \left( \begin{array}{c} a_{p+1,p} \\ a_{p+2,p} \\ \vdots \\ a_{n,p} \end{array} \right)$$

the rotation itself can be calculated as follows:

$$\begin{eqnarray*}\sigma &=& \boldsymbol{x}\cdot\boldsymbol{x} \\ \vert\boldsym... ...t \boldsymbol{Q}_{j+1} \\ \boldsymbol{Q} &=& \boldsymbol{Q}_1 \end{eqnarray*}$$