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Subsection 10.3.3 Givens' 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*}
Remark 10.3.3.1.

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.