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Suppose

  • $P_\Delta(\lambda ) = (A_m + \Delta_m) \lambda ^m + \cdots + (A_1 + \Delta _1)\lambda ^1 + (A_0 + \Delta _0)$ is a matrix polynomial, and $\lambda $ is a complex variable.
  • $A_j,\Delta _j \in \mathbb{C}^{n \times n},\quad (j = 0,1,2,\ldots,m)$

  • $w_j \ge 0,(j = 0,1,\ldots,m)$ and $w_0>0$.

  • $A = \left\{ P_\Delta(\lambda ):\left\| \Delta_j \right\| \le \varepsilon w_j, j = 0,1,2,\ldots,m \right\}$

  • $\| \cdot \|$ is subordinate matrix norm, and $\partial A$ is boundary of $A$.

  • The open ball of radius $δ$ about $P^{(0)}_{\Delta}(\lambda)$, is $\{P_{\Delta}(\lambda): \|\Delta_j - \Delta_j^{(0)}\| < \delta \text{ for all } j \}$.

With hypotheses, why does;

$\partial A = \{ P_\Delta (\lambda ):\|\Delta_j\| \le \varepsilon w_j$ for each $j $, and equality holds for at least one $j$ $\}$ ?

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