Let $A\in \mathbb{R^{\textit{k}\times \textit{k}}}$ a matrix with superior Hessenberg shape. This prove justify the use of $\rho = a_{\textit{kk}}$ as the dinamic shift of the $QR$ algorithm.
Show that $a_{\textit{kk}}$ is the a Rayleigh quotient associated to the canonical vector $e_k=(0,...,0,1)$
We want $a_{\textit{kk}} = \frac{e_k^TAe_k}{e_k^Te_k}$
Any hints of how to beggin?
Hint: It is generally true that $e_i^TAe_k = a_{ij}$. To prove this, I would recommend that you first convince yourself that it works for the $3 \times 3$ case, then prove it in general using your preferred definition of matrix multiplication.