What is happening in this proof?

Question

I’m trying to understand the proof of Schur’s theorem, but I’m having trouble with one line. I don’t understand equation 1, and the authors I have looked at seem to think it’s obvious because none of them explain it. I even tried calculating an example but this did not help. What’s going on?

Answer 1

The equation is equivalent to $$AQ_1=Q_1\pmatrix{\lambda_1&*\\0&*}.$$ These matrices will be the same if their first columns are the same. But as $q_1$ is the first column of $Q_1$, their first columns are $Aq_1=\lambda q_1$.