# Generalization of QR factorization

Find the QR factorization of the general $2 \times 2$ matrix $A = \begin{bmatrix}a & b\\c & d\end{bmatrix}$?

• Not sure what you expect. A QR factorization always exists. – Martin Argerami Oct 30 '16 at 18:36
• A=QR. How can I express QR in terms of a,b,c,d? – user7416 Oct 30 '16 at 18:37
• @MartinArgerami edited the question – user7416 Oct 30 '16 at 18:39

It would have probably been easier to ask Wolfram Alpha than Math.SE. You have $A=QR$, where $$Q=\begin{bmatrix} \dfrac{\bar a}{\sqrt{|a|^2+|b|^2}}&\dfrac{\bar c}{\sqrt{|a|^2+|b|^2}}\\ \frac{\bar b|c|^2-c\bar a\bar d}{\sqrt{(|a|^2+|c|^2)(|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c)}} &\dfrac{\bar d|a|^2-a\bar b\bar c}{\sqrt{(|a|^2+|c|^2)(|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c)}} \end{bmatrix},$$
$$R=\begin{bmatrix} {\sqrt{|a|^2+|b|^2}}&\dfrac{b\bar a-d\bar c}{\sqrt{|a|^2+|b|^2}}\\ 0&\dfrac{\bar d|a|^2-a\bar b\bar c}{\sqrt{\frac{|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c}{|a|^2+|c|^2}}} \end{bmatrix}$$