Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 2 / 18C / Solution

Choose the Givens rotation convention in which ${\Omega }^{[p,q]}$ equals the identity outside rows and columns $p,q$, and has blockin those coordinates. It is an orthogonal matrix. For any chosen column $j$, write $u=A_{pj}$, $v=A_{qj}$ and $r=\sqrt{u^2+v^2}$. If $r>0$, take $\cos \theta =u/r$, $\sin \theta =v/r$. Then the new entries are $r$ and $-vu/r+uv/r=0$. If both entries vanish, take $\theta =0$. This proves the requested elimination for each chosen $j$; the angle is allowed to depend on $j$, and one angle need not eliminate an entire arbitrary row.

For the given matrix, use $\cos \theta =\sin \theta =1/\sqrt{2}$ first for ${\Omega }^{[1,2]}$, obtainingUse the same cosine and sine for ${\Omega }^{[1,3]}$. It makesThis is already upper triangular, including the zero $(3,2)$ entry, and every leading nonzero row entry is positive. Thus the requested QR decomposition isThe orthogonal matrix property follows from the two rotations, and direct multiplication verifies $A=QR$.