General formula for $QR$ decomposition in two by two case.

Question

Let $A = \left( \begin{matrix} a & b \\ c & d \end{matrix} \right)$ be a two by two matrix over the field of complex numbers $\mathcal{C}$. Are there some reference which gives the decomposition $A = QR$, where $Q$ is a unitary matrix and $R$ is an upper triangular matrix? Thank you very much.

Answer 1

LAPACK Working Note 148 gives an expression for the complex Givens rotation $\mathbf Q^\ast=\begin{pmatrix}\chi&\sigma\\-\bar{\sigma}&\chi\end{pmatrix}$ such that

$$\mathbf Q^\ast\begin{pmatrix}a\\c\end{pmatrix}=\begin{pmatrix}r\\0\end{pmatrix}$$

where $\chi^2+|\sigma|^2=1$.

The paper then derives the expressions

$$\begin{align*}\chi&=\frac{|a|}{\sqrt{|a|^2+|c|^2}}\\\sigma&=\frac{\operatorname{sign}(a)\,\bar{c}}{\sqrt{|a|^2+|c|^2}}\\\end{align*}$$

where $\operatorname{sign}(a)=\begin{cases}\frac{a}{|a|}&a\ne0\\1&a=0\end{cases}$.

From that, you can now derive the expression for $\mathbf R$ by premultiplying $\mathbf A$ with $\mathbf Q^\ast$.