I was reading a proof about the triangularization theorem given in Shapiro's Linear Algebra and Matrices textbook. Here is the proof for reference
I understand the basic gist of the proof, however, there is one thing that doesn't sit right with me. How are we able to justify that after constructing the invertible matrix $W$ as described, that $W^{-1} A W$ will take the form $$\begin{pmatrix} \lambda & * \\ 0 & A_{22} \: \end{pmatrix}$$ Any help will be appreciated.