Consider a function
\begin{equation}
Q^{(p)}(\alpha) := Q_p + \alpha (Q_{p-1} - Q_p), \quad \alpha \in [0,1]
\end{equation} where $Q_p, Q_{p-1}$ are real matrices in $\mathbb{R}^{n \times n}$.
The question is: at what point $\alpha$ the norm $Q^{(p)}(\alpha)$ achieves its maximum value?
I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$.
I am unable to prove it.
So I considered an easier version of the problem.
Let $M=(p, q)$ be an ideal in $\mathbb C[x, y]$, where $p$ and $q$ are elem...
