So I have a function $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ given by $$f(x,y) = \sum_{i=1}^m y_i A_ix$$ where $A_i$ are $n$ by $n$ real symmetric positive definite matrices (not that it really matters in this case) for $i=1, \dots m$.
I would like to calculate the Lipschitz constant of $\nabla_x f(x,y)$ on the set that $\|x \| \leq 1$. When I fix $x \in \mathbb{R}^n\cap\{x: \|x\| \leq 1\}$ and take $y^1,y^2 \in \mathbb{R}^m$ I have
$$ \|\nabla_x f(x,y^1) - \nabla_x f(x,y^2)\| = \|\sum_{i=1}^m (y^1_i - y^2_i) A_ix\| $$ but I'm having trouble majorizing this for $L\|y^1 - y^2\|$. How can I calculate such an $L$ so that the inequality works?