Past exam of the numerical analysis course of the University of Oxford / 2015 / 3 / d / i / Solution

Apply the product from right to left, choosing every Givens rotation from the current first column. First $J(1,2)$ replaces $(a_1,a_2)$ by $(\sqrt{a_1^2+a_2^2},0)$. Next $J(1,3)$ combines that retained entry with $a_3$ and sets the third entry to zero. Inductively, after eliminating coordinates $2,\dots ,k$, the first coordinate isand coordinates $2,\dots ,k$ are zero. Later Givens rotations involve only coordinate one and a new coordinate, so those zeros remain zero. Thereforewhere $e_1$ is the first standard basis vector. The zero vector is included by choosing the identity matrix for every zero pair. The nonnegative sign follows from the convention fixed in part (c)(ii). This is the first-column stage of QR decomposition by Givens rotations.