Smallest Lipschitz constant of $x \mapsto A_0 + \sum_{i=1}^n x_i A_i$

Question

Let $\mathcal{S}_m (\mathbb{R})$ denote the space of symmetric $m \times m$ matrices.

Consider the matrix-valued affine-linear function $A: \mathbb{R}^n \to \mathcal{S}_m (\mathbb{R})$ given by

$$ A(x) := A_0 + \sum_{i=1}^{n} x_i A_i $$

with $A_0, A_1, \dots , A_n \in \mathcal{S}_m (\mathbb{R})$.

Function $A: \mathbb{R}^n \to \mathcal{S}_m (\mathbb{R})$ is affine-linear and thus (globally) Lipschitz-continuous.

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Question: What is the (smallest) Lipschitz-constant $L>0$ of $A: \mathbb{R}^n \to \mathcal{S}_m (\mathbb{R})$ such that

$$ \| A(y) - A(x) \|_2 \leq L \| y -x \|_2 \qquad \forall x,y \in \mathbb{R}^n$$

where $\| z \|_2 = \sqrt{z^{\top}z}$ for any vector $z \in \mathbb{R}^n$ and

$$\| B \|_2 = \sup_{\| x\| =1} \| Bx \|_{2} \qquad \forall B \in \mathbb{R}^{m \times m} $$

denotes the induced matrix norm.

Answer 1

_{The following is a very rough sketch written in haste. It may be totally wrong. Please check every step very carefully.}

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$$ {\rm A} ({\rm y}) - {\rm A} ({\rm x}) = \sum_{k=1}^n \left( y_k - x_k \right) {\rm A}_k = \begin{bmatrix} {\rm A}_1 & {\rm A}_2 & \cdots & {\rm A}_n\end{bmatrix} \left( ({\rm y} - {\rm x}) \otimes {\rm I}_m \right) $$

where $\otimes$ denotes the Kronecker product. Since the spectral norm is submultiplicative,

$$\begin{aligned} \left\| {\rm A} ({\rm y}) - {\rm A} ({\rm x}) \right\|_2 &= \left\| \begin{bmatrix} {\rm A}_1 & {\rm A}_2 & \cdots & {\rm A}_n\end{bmatrix} \left( ({\rm y} - {\rm x}) \otimes {\rm I}_m \right) \right\|_2 \\\\ &\leq \left\| \begin{bmatrix} {\rm A}_1 & {\rm A}_2 & \cdots & {\rm A}_n\end{bmatrix} \right\|_2 \underbrace{\left\| ({\rm y} - {\rm x}) \otimes {\rm I}_m \right\|_2}_{=: \| {\rm y} - {\rm x} \|_2 }\end{aligned}$$