Subsection10.3.3Givens' rotations

First, we introduce another important class of unitary matrices known as Givens' rotations. Given a vector $x = \left( \begin{array}{c} \chi_1 \\ \chi_2 \end{array} \right) \in \R^2\text{,}$ there exists an orthogonal matrix $G$ such that $G^T x = \left( \begin{array}{c} \pm \| x \|_2 \\ 0 \end{array} \right) \text{.}$ The Householder transformation is one example of such a matrix $G\text{.}$ An alternative is the Givens' rotation: $G = \left( \begin{array}{c c} \gamma \amp -\sigma \\ \sigma \amp \gamma \end{array} \right)$ where $\gamma^2 + \sigma^2 = 1 \text{.}$ (Notice that $\gamma$ and $\sigma$ can be thought of as the cosine and sine of an angle.) Then

\begin{equation*} \begin{array}{rcl} G^T G \amp= \amp \left( \begin{array}{c c} \gamma \amp -\sigma \\ \sigma \amp \gamma \end{array} \right)^T \left( \begin{array}{c c} \gamma \amp -\sigma \\ \sigma \amp \gamma \end{array} \right) = \left( \begin{array}{c c} \gamma \amp \sigma \\ -\sigma \amp \gamma \end{array} \right) \left( \begin{array}{c c} \gamma \amp -\sigma \\ \sigma \amp \gamma \end{array} \right) \\ \amp= \amp \left( \begin{array}{c c} \gamma^2 + \sigma^2 \amp - \gamma \sigma + \gamma \sigma \\ \gamma\sigma - \gamma\sigma \amp \gamma^2 + \sigma^2 \end{array} \right) = \left( \begin{array}{c c} 1 \amp 0 \\ 0 \amp 1 \end{array} \right), \end{array} \end{equation*}

which means that a Givens' rotation is a unitary matrix.

Now, if $\gamma = \chi_1 / \| x \|_2$ and $\sigma = \chi_2 / \| x \|_2 \text{,}$ then $\gamma^2 + \sigma^2 = ( \chi_1^2 + \chi_2^2 ) / \| x \|_2^2 = 1$ and

\begin{equation*} \left( \begin{array}{c c} \gamma \amp -\sigma \\ \sigma \amp \gamma \end{array} \right)^T \left( \begin{array}{c} \chi_1 \\ \chi_2 \end{array} \right) = \left( \begin{array}{c c} \gamma \amp \sigma \\ -\sigma \amp \gamma \end{array} \right) \left( \begin{array}{c} \chi_1 \\ \chi_2 \end{array} \right) = \left( \begin{array}{c} ( \chi_1^2 + \chi_2^2 ) / \| x \|_2 \\ ( \chi_1 \chi_2 - \chi_1 \chi_2 ) / \| x \|_2 \end{array} \right) = \left( \begin{array}{c} \| x \|_2 \\ 0 \end{array} \right). \end{equation*}
Remark10.3.3.1.

We only discuss real valued Givens' rotations and how that transform real valued vectors since the output of our reduction to tridiagonal form yields a real valued tridiagonal matrix.