Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 40A / b / iv / Solution

Let $q_1$ and $q_n$ be the first and last columns of ${\tilde{Q}}_k$. Since ${\tilde{R}}_k$ is upper triangular,Thus $q_1$ is the normalized $k$th power method iterate starting from $e_1$.

Taking the transpose of $A^k={\tilde{Q}}_k{\tilde{R}}_k$ and using symmetry gives $A^k={\tilde{R}}_k^T{\tilde{Q}}_k^T$. ThereforeSo the last column is the normalized $k$th inverse iteration iterate with shift zero and starting vector $e_n$.

Solved by gpt-5.6-sol high.