NLA MT25, Givens rotations

$G(i, j, \theta _ {ij})$ is the $m \times m$ identity matrix except for four entries: the $(i,i)$ and $(j,j)$ entries equal $\cos \theta _ {ij}$, the $(i,j)$ entry equals $-\sin \theta _ {ij}$, and the $(j,i)$ entry equals $\sin \theta _ {ij}$:

\[G(i, j, \theta _ {ij}) = \begin{bmatrix} I & & & & \\ & \cos\theta _ {ij} & & -\sin\theta _ {ij} & \\ & & I & & \\ & \sin\theta _ {ij} & & \cos\theta _ {ij} & \\ & & & & I \end{bmatrix}\]

where the four displayed scalars sit at rows/columns $i$ and $j$ and the $I$ blocks are identity.

It is orthogonal ($\cos^2\theta _ {ij} + \sin^2\theta _ {ij} = 1$, so $G^\ast G = I$) but not symmetric, so $G^{-1} = G^\ast \ne G$ — unlike a Householder reflector. It acts locally: left-multiplication modifies only rows $i$ and $j$, right-multiplication only columns $i$ and $j$.

When used to triangularise a matrix, $\theta _ {ij}$ is chosen so that left-multiplication by $G(i, j, \theta _ {ij})$ makes the $(i,j)$ entry of the product zero. Concretely, to zero an entry $b$ sitting below a pivot $a$, solving

\[\begin{bmatrix} c & -s \\ s & c \end{bmatrix}\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} r \\ 0 \end{bmatrix}, \qquad r = \sqrt{a^2 + b^2}\]

gives $\cos\theta _ {ij} = a/r$ and $\sin\theta _ {ij} = -b/r$.