Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 4 / 6C / Solution

A Givens rotation is identity except for a $2\times 2$ block $\left(\begin{array}{cc}c& s\\ -s& c\end{array}\right)$ in rows $p,q$. Choose $p=j,q=i$ and $c=a_{jj}/r$, $s=a_{ij}/r$, $r=(a_{jj}^2+a_{ij}^2{)}^{1/2}$, with sign adjusted to the block convention; this mixes only those rows and zeros $a_{ij}$. For $3\times 3$, successively apply rotations $(1,2)$, $(1,3)$, $(2,3)$, producing patterns $\ast \ast \ast ;0\ast \ast ;\ast \ast \ast$, then $\ast \ast \ast ;0\ast \ast ;0\ast \ast$, then an upper triangular $R$. The product of transposed rotations is $Q$.