Question

Find the QR factorization of $\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}0& -3& 0\\ 0& 0& 0\\ 2& 0& 0\\ 0& 0& 4\end{array}\right]$.

Short answer

The solution is $\mathrm{QR}=\left[\begin{array}{ccc}0& -1& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$.

Step-by-step solution

`Explanation of the solution

Consider the matrix A as follows.

$\mathrm{A}=\left[\begin{array}{ccc}0& -3& 0\\ 0& 0& 0\\ 2& 0& 0\\ 0& 0& 4\end{array}\right]$

The vectors are orthogonal.

role="math" localid="1660103522754" $$\begin{gathered} {\stackrel{\rightharpoonup }{\mathrm{V}}}_1=2{\stackrel{\rightharpoonup }{\mathrm{e}}}_3 \\ {\stackrel{\rightharpoonup }{\mathrm{V}}}_2=-3{\stackrel{\rightharpoonup }{\mathrm{e}}}_1 \\ {\stackrel{\rightharpoonup }{\mathrm{V}}}_3=4{\stackrel{\rightharpoonup }{\mathrm{e}}}_4 \\ \mathrm{Q}=\left[\begin{array}{cc}\frac{{\stackrel{\rightharpoonup }{\mathrm{V}}}_1}{||{\stackrel{\rightharpoonup }{\mathrm{V}}}_1||}& \frac{{\stackrel{\rightharpoonup }{\mathrm{V}}}_2}{||{\stackrel{\rightharpoonup }{\mathrm{V}}}_2||}\end{array}\right] \\ \begin{array}{cc}\frac{{\stackrel{\rightharpoonup }{\mathrm{V}}}_3}{||{\stackrel{\rightharpoonup }{\mathrm{V}}}_3||}=\left[\begin{array}{ccc}0& -1& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]& \end{array} \\ \mathrm{R}=\left[\begin{array}{ccc}||{\stackrel{\rightharpoonup }{\mathrm{V}}}_1||& 0& 0\\ 0& ||{\stackrel{\rightharpoonup }{\mathrm{V}}}_2||& 0\\ 0& 0& ||{\stackrel{\rightharpoonup }{\mathrm{V}}}_3||\end{array}\right] \\ \mathrm{R}=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right] \\ \mathrm{Thus},\mathrm{the}\mathrm{QR}\mathrm{factorization}\mathrm{is}\mathrm{QR}=\left[\begin{array}{ccc}0& -1& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]. \end{gathered}$$