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

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  • $\begingroup$ Not sure what you expect. A QR factorization always exists. $\endgroup$ – Martin Argerami Oct 30 '16 at 18:36
  • $\begingroup$ A=QR. How can I express QR in terms of a,b,c,d? $\endgroup$ – user7416 Oct 30 '16 at 18:37
  • $\begingroup$ @MartinArgerami edited the question $\endgroup$ – user7416 Oct 30 '16 at 18:39
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It would have probably been easier to ask Wolfram Alpha than Math.SE. You have $A=QR$, where $$ Q=\begin{bmatrix} \dfrac{\bar a}{\sqrt{|a|^2+|b|^2}}&\dfrac{\bar c}{\sqrt{|a|^2+|b|^2}}\\ \frac{\bar b|c|^2-c\bar a\bar d}{\sqrt{(|a|^2+|c|^2)(|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c)}} &\dfrac{\bar d|a|^2-a\bar b\bar c}{\sqrt{(|a|^2+|c|^2)(|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c)}} \end{bmatrix}, $$

$$ R=\begin{bmatrix} {\sqrt{|a|^2+|b|^2}}&\dfrac{b\bar a-d\bar c}{\sqrt{|a|^2+|b|^2}}\\ 0&\dfrac{\bar d|a|^2-a\bar b\bar c}{\sqrt{\frac{|b|^2|c|^2+|a|^2|d|^2-2\text{Re}\,ad\bar b\bar c}{|a|^2+|c|^2}}} \end{bmatrix} $$

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    $\begingroup$ Thank you Martin, however I'm looking for the steps not just the final result. $\endgroup$ – user7416 Oct 30 '16 at 19:01
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    $\begingroup$ @user1023 The steps are just Gram-Schmidt, being careful about the singular case. $\endgroup$ – Ian Oct 30 '16 at 19:33
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https://www.youtube.com/watch?v=51MRHjKSbtk This link was super helpful for anyone looking for the answer

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