矩阵多项式特征值估计在现代工程学和数学物理等很多领域有重要的应用^[1]. 矩阵多项式特征值估计方法往往可以在一般矩阵特征值估计方法的基础上, 通过适当的推广而得到, 而矩阵特征值估计方法较为成熟. Brauer A在1947年通过改进Gershgorin定理, 给出了矩阵特征值估计的Brauer定理^[2]. Li Chaoqian和Li Yaotang在文[3] 中推广了矩阵特征值估计的Brauer定理, 给出了更为精确的广义Brauer集. 而矩阵多项式特征值的精确计算较为困难, 关于其数值计算见[4, 5]. 在一些应用中, 需要对矩阵多项式特征值的界进行估计. 2001年, Tisseur F和Higham N J在文[6]中刻画了矩阵多项式的伪谱, 以此来估计其特征值. 2009年, 王学锋, 王卫国和刘新国介绍了几类特殊类型的矩阵多项式特征值界的估计方法^[7]. 2018年, Michailidou C, Panayiotis P将矩阵特征值估计的Gershgorin定理, 广义Gershgorin定理, Brauer定理, Dashnic-Zusmanovich定理进行推广, 得到了矩阵多项式的特征值估计相关定理以及相应的结果^[8]. 本文在以上工作的基础上, 将矩阵特征值估计的广义Brauer集推广到矩阵多项式, 给出了其特征值估计的广义Brauer集, 并对该集合的性质进行了详细的研究.

本文主要研究矩阵多项式$ P(\lambda) $, 即

$ \begin{equation} P(\lambda)=A_{m}\lambda^{m}+A_{m-1}\lambda^{m-1}+\cdots+A_{1}\lambda+A_{0}, \end{equation} $

(1.1)

其中$ A_{0}, A_{1}, \cdots, A_{m}\in\mathbb{C}^{n\times n}\; (A_{m}\neq0), $ $ \lambda $为一个复变量, 且行列式det $ P(\lambda) $不完全为零. 若$ x_{0}\; (x_{0}\in\mathbb{C}^{n}) $是$ P(\mu)\; x=0 $的非零解, 则称$ \mu $为$ P(\lambda) $的特征值, 且$ x_{0} $是对应于$ \mu $的特征向量. $ P(\lambda) $的有限谱是所有有限特征值构成的集合, 即

$ \begin{equation*} \sigma(P)=\{\mu\in\mathbb{C}:\mathrm{det}(P(\mu))=0\}=\{\mu\in\mathbb{C}:0\in\sigma(P(\mu))\}. \end{equation*} $

此外, 对于无穷特征值$ \mu=\infty $, 考虑矩阵多项式

$ \begin{equation*} \label{eq:2}\hat{P}(\lambda):=\lambda^{m}P(1/\lambda)=A_{0}\lambda^{m}+A_{1}\lambda^{m-1}+\cdots+A_{m-1}\lambda+A_{m}, \end{equation*} $

可验证$ 0 $是$ \hat{P}(\lambda) $的特征值当且仅当$ \infty $是$ P(\lambda) $的特征值, 即可转化为研究$ \hat{P}(\lambda) $的$ 0 $特征值问题.

下面列出与本文相关的引理和符号.

全文中规定$ N=\{1, 2, \cdots, n\} $.

引理1.1^[3]  设矩阵$ A\in\mathbb{C}^{n\times n} $, 对于任意的$ i, \; j\in N, \; j<i $, $ \alpha\in[0, 1] $, 记

$ \begin{equation} \mathcal{T}_{i, j, \alpha}(A):=\{\mu\in \mathbb{C}:\mid\mu-(A)_{i, i}\mid\mid\mu-(A)_{j, j}\mid \leq\ \alpha r_i(A)r_j(A)+(1-\alpha)c_i(A)c_j(A)\}, \\ \end{equation} $

(1.2)

则

$ \begin{equation*} \sigma(A)\subseteq\mathcal{T}_{\alpha}(A):=\bigcup\limits_{i\in N} \bigcup\limits^{i-1}_{j=1} \mathcal{T}_{i, j, \alpha}(A), \end{equation*} $

进而

$ \begin{equation} \sigma(A)\subseteq\mathcal{T}(A):=\bigcap\limits_{\alpha\in[0, 1]}\mathcal{T}_{\alpha}(A).\\ \end{equation} $

(1.3)

其中$ r_i(A)=\sum\limits_{j\in N\setminus\{i\}}\mid(A)_{i, j}\mid, \; c_i(A)=r_i(A^T), $ $ (A)_{i, j} $为$ A $的第$ (i, \; j) $项. 称$ \mathcal{T}_{i, j, \alpha}(A) $为矩阵$ A $的第$ (i, \; j) $项广义Brauer集, 称$ \mathcal{T}(A) $为矩阵$ A $的广义Brauer集.

类似于$ (1.2) $式, 对于任意的$ i, \; j\in N, \; j<i $, $ \alpha\in[0, 1] $, 下面给出矩阵多项式$ P(\lambda) $的相关集合. 记

$ \begin{equation} \begin{split} &\mathcal{T}_{i, j, \alpha}(P) :=\{\mu\in \mathbb{C}:0\in\mathcal{T}_{i, j, \alpha}(P(\mu))\}=\\ &\{\mu\in \mathbb{C}:|(P(\mu))_{i, i}||(P(\mu))_{j, j}|\leq \alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu))\}, \end{split} \end{equation} $

(1.4)

$ \begin{equation*} \mathcal{T}_{\alpha}(P):=\{\mu\in \mathbb{C}:0\in\mathcal{T}_{\alpha}(P(\mu))\}=\bigcup\limits_{i\in\, N}\bigcup\limits^{i-1}_{j=1}\mathcal{T}_{i, j, \alpha}(P), \end{equation*} $

$ \begin{equation} \mathcal{T}(P):=\{\mu\in \mathbb{C}:0\in\mathcal{T}(P(\mu))\}=\bigcap\limits_{\alpha\in[0, 1]}\mathcal{T}_{\alpha}(P).\\ \end{equation} $

(1.5)

其中$ r_i(P(\mu))=\sum\limits_{j\in N\setminus\{i\}}\mid(P(\mu))_{i, j}\mid, \; c_i(P(\mu))=r_i((P(\mu))^T), $同时称$ \mathcal{T}_{i, j, \alpha}(P) $为矩阵多项式$ P(\lambda) $的第$ (i, \; j) $项广义Brauer集, 称$ \mathcal{T}(P) $为矩阵多项式$ P(\lambda) $的广义Brauer集. 此外,

$ \begin{equation} \begin{split} \mathcal{T}_{i, j, \alpha}(\hat{P})\backslash\{0\} :=\{\mu\in\mathbb{C}\backslash\{0\}:0\in\mathcal{T}_{i, j, \alpha}(P(\mu^{-1}))\}=\{\mu\in\mathbb{C}\backslash\{0\}:\mu^{-1}\in \mathcal{T}_{i, j, \alpha}(P)\}, \end{split} \end{equation} $

(1.6)

进一步

$ \begin{equation*} \begin{split} &\mathcal{T}(\hat{P})\backslash\{0\} :=\{\mu\in\mathbb{C}\backslash\{0\}:0\in\mathcal{T}(P(\mu^{-1}))\}=\{\mu\in\mathbb{C}\backslash\{0\}:\mu^{-1}\in\mathcal{T}(P)\}, \end{split} \end{equation*} $

即当$ \mu=0 $位于$ \mathcal{T}_{i, j, \alpha}(\hat{P}) $中(也位于$ \mathcal{T}(\hat{P}) $中)时, $ \mu=\infty $位于$ \mathcal{T}_{i, j, \alpha}(P) $中(也位于$ \mathcal{T}(P) $中).

在$ (1.4) $式中, 当$ \alpha=1 $和$ 0 $时, 集合退化为文献[8]中定义$ 4.1 $的Brauer集及转置情形的Brauer集,

$ \begin{equation*} \begin{split} \mathcal{B}_{i, j}(P) :=&\{\mu\in \mathbb{C}:0\in \mathcal{B}_{i, j}(P(\mu))\}\\ =&\{\mu\in \mathbb{C}:|(P(\mu))_{i, i}||(P(\mu))_{j, j}|\leq r_i(P(\mu))r_j(P(\mu))\}, \\ \mathcal{B}(P):=&\{\mu\in \mathbb{C}:0\in \mathcal{B}(P(\mu))\}=\bigcup\limits_{i\in N}\bigcup\limits^{i-1}_{j=1}\mathcal{B}_{i, j}(P).\\ \end{split} \end{equation*} $

因此本文只讨论$ \alpha\in(0, 1) $的情况.

下面利用矩阵多项式的广义Brauer估计其特征值的范围.

定理2.1  设$ P(\lambda) $为由$ (1.1) $式定义的矩阵多项式, 则$ P(\lambda) $的特征值包含在其广义Brauer集($ (1.5) $式) 中.

证  设$ \mu $为矩阵多项式$ P(\lambda) $的有限特征值, 即$ \mu\in\sigma(P) $, 由$ (1.3) $式有$ 0\in\sigma(P(\mu))\subseteq\mathcal{T}(P(\mu)) $, 进而有$ \mu\in\mathcal{T}(P) $. 对于$ P(\lambda) $的无限特征值$ \mu=\infty $, 有$ 0\in\sigma(\hat{P}(\mu))\subseteq\mathcal{T}(\hat{P}(\mu)), $即$ \mu=\infty $包含在$ \mathcal{T}(P) $中. 故广义Brauer集包含$ P(\lambda) $的所有特征值.

接着讨论矩阵多项式的广义Brauer集的基本性质.

定理2.2  令$ i, \; j\in N $, 则对由$ (1.1) $式中定义的$ n\times n $阶矩阵多项式$ P(\lambda) $, 以下结论成立

(ⅰ) $ \mathcal{T}_{i, j, \alpha}(P) $是$ \mathbb{C} $的闭子集.

(ⅱ) 对任意$ b\in\mathbb{C}\backslash\{0\}, $若矩阵多项式满足

$ Q_1(\lambda)=P(b\lambda), \; Q_2(\lambda)=P(\lambda+b), $

则有

$ \mathcal{T}_{i, j, \alpha}(Q_1)=b^{-1}\mathcal{T}_{i, j, \alpha}(P), \; \mathcal{T}_{i, j, \alpha}(Q_2)=\mathcal{T}_{i, j, \alpha}(P)-b. $

(ⅲ) 如果$ (P(\lambda))_{i, i}=0 $或$ (P(\lambda))_{j, j}=0 $, 那么$ \mathcal{T}_{i, j, \alpha}(P)=\mathbb{C}. $

(ⅳ) 若系数矩阵$ A_0, \; A_1, \; \cdots, \; A_m $的第$ i $行, 第$ i $列; 第$ j $行, 第$ j $列均为实数, 则$ \mathcal{T}_{i, j, a}(P) $相对于实轴对称.

证  (ⅰ) 令$ \mu\not\in\mathcal{T}_{i, j, \alpha}(P), $于是

$ |(P(\mu))_{i, i}||(P(\mu))_{j, j}|>\alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu)). $

由连续性, 存在无限接近于$ \mu $的$ \hat{\mu} $, 使得

$ |(P(\hat{\mu}))_{i, i}||(P(\hat{\mu}))_{j, j}|> \alpha r_i(P(\hat{\mu}))r_j(P(\hat{\mu}))+(1-\alpha)c_i(P(\hat{\mu}))c_j(P(\hat{\mu})), $

故$ \hat{\mu}\not\in\mathcal{T}_{i, j, \alpha}(P), $因此集合$ \mathbb{C}\backslash\mathcal{T}_{i, j, \alpha}(P) $是开集, 即$ \mathcal{T}_{i, j, \alpha}(P) $是$ \mathbb{C} $的闭子集.

(ⅱ) 通过$ (1.4) $式, 易得

$ \begin{equation*} \begin{split} &\mathcal{T}_{i, j, \alpha}(Q_1)=\{\mu\in \mathbb{C}:0\in\mathcal{T}_{i, j, \alpha}(P(b\mu))\}=\{{\frac{\mu}{b}}\in \mathbb{C}:0\in\mathcal{T}_{i, j, \alpha}(P(\mu))\}, \\ &\mathcal{T}_{i, j, \alpha}(Q_2)=\{\mu\in \mathbb{C}:0\in\mathcal{T}_{i, j, \alpha}(P(\mu+b))\}=\{\mu-b\in \mathbb{C}:0\in\mathcal{T}_{i, j, \alpha}(P(\mu))\}. \end{split} \end{equation*} $

(ⅲ) 如果$ (P(\lambda))_{i, i}=0 $或$ (P(\lambda))_{j, j}=0 $, 那么对任意的$ \mu\in\mathbb{C} $, 不等式

$ |(P(\mu))_{i, i}||(P(\mu))_{j, j}|\leq \alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu)) $

恒成立, 于是$ \mathcal{T}_{i, j, \alpha}(P)=\mathbb{C}. $

(ⅳ) 对任意的$ \mu\in\mathcal{T}_{i, j, \alpha}(P), $如果系数矩阵$ A_0, \; A_1, \cdots, \; A_m $的第$ i $行, 第$ i $列; 第$ j $行, 第$ j $列均为实数, 那么

$ \begin{equation*} \begin{split} \left|\sum\limits^m_{k=0}(A_k)_{i, i}\mu^k\right|\left|\sum\limits^m_{k=0}(A_k)_{j, j}\mu^k\right| \leq \alpha&\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\mu^k\right| \right)\cdot \left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\mu^k\right|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\mu^k\right|\right)\cdot \left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\mu^k\right|\right), \end{split} \end{equation*} $

因此

$ \begin{equation*} \begin{split} \left|\overline{\sum\limits^m_{k=0}(A_k)_{i, i}\mu^k}\right|\left|\overline{\sum\limits^m_{k=0}(A_k)_{j, j}\mu^k}\right| \leq \alpha&\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\overline{\sum\limits^m_{k=0}(A_k)_{i, p}\mu^k}\right|\right)\cdot\left(\sum\limits_{p\in N\backslash\{j\}}\left|\overline{\sum\limits^m_{k=0}(A_k)_{j, p}\mu^k}\right|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\overline{\sum\limits^m_{k=0}(A_k)_{q, i}\mu^k}\right|\right)\cdot \left(\sum\limits_{q\in N\backslash\{j\}}\left|\overline{\sum\limits^m_{k=0}(A_k)_{q, j}\mu^k}\right|\right) \end{split} \end{equation*} $

或

$ \begin{equation*} \begin{split} \left|\sum\limits^m_{k=0}(A_k)_{i, i}\overline{\mu}^k\right|\left|\sum\limits^m_{k=0}(A_k)_{j, j}\overline{\mu}^k\right| \leq \alpha&\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\overline{\mu}^k\right|\right)\cdot\left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\overline{\mu}^k\right|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\overline{\mu}^k\right|\right)\cdot \left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\overline{\mu}^k\right|\right). \end{split} \end{equation*} $

于是$ \bar{\mu}\in\mathcal{T}_{i, j, \alpha}(P), $故$ \mathcal{T}_{i, j, \alpha}(P) $相对于实轴对称.

在$ (1.4) $式中, 当不等式分别取“$ < $”和“=”时, 广义Brauer中内点和边界点满足如下的性质.

定理2.3  令$ i, \; j\in N, \; j<i $, 那么集合

$ \{\mu\in\mathbb{C}:|(P(\mu))_{i, i}||(P(\mu))_{j, j}|<\alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu))\} $

位于$ \mathcal{T}_{i, j, \alpha}(P) $集的内部. 进一步, $ \mathcal{T}_{i, j, \alpha}(P) $的边界$ \partial\mathcal{T}_{i, j, \alpha}(P) $满足

$ \begin{equation*} \begin{split} \partial\mathcal{T}_{i, j, \alpha}(P)\subseteq\{\mu\in \mathbb{C}:0\in \partial\mathcal{T}_{i, j, \alpha}(P(\mu))\} =&\{\mu\in \mathbb{C}:|(P(\mu))_{i, i}||(P(\mu))_{j, j}|\\ =& \alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu))\}. \end{split} \end{equation*} $

证  设$ \mu\in\mathcal{T}_{i, j, \alpha}(P), $即

$ \begin{equation*} \begin{split} |(P(\mu))_{i, i}||(P(\mu))_{j, j}|< \alpha r_i(P(\mu))r_j(P(\mu))+(1-\alpha)c_i(P(\mu))c_j(P(\mu)). \end{split} \end{equation*} $

由连续性, 存在$ \varepsilon>0, $对于$ \hat{\mu}\in\mathbb{C} $, 当$ |\mu-\hat{\mu}|\leq\varepsilon $时, 都有

$ \begin{equation*} \begin{split} |(P(\hat{\mu}))_{i, i}||(P(\hat{\mu}))_{j, j}|< \alpha r_i(P(\hat{\mu}))r_j(P(\hat{\mu}))+(1-\alpha)c_i(P(\hat{\mu}))c_j(P(\hat{\mu})), \end{split} \end{equation*} $

于是$ \mu $是$ \mathcal{T}_{i, j, \alpha}(P) $的内点.

下面再讨论广义Brauer的有界性. 对于$ i, \; j\in N, \; j<i $, 定义如下集合

$\beta_{i}=\{p\in \mathcal{N}:(A_{m})_{i, p}\neq 0\}, \; \; \; \; \overline{\beta}_{i}=N\setminus \beta_{i}=\{p\in \mathcal{N}:(A_{m})_{i, p}=0\}, \\ \gamma_{i}\; =\{q\in \mathcal{N}:(A_{m})_{q, i}\neq 0\}, \; \; \; \; \overline{\gamma}_{i}=N\setminus \gamma_{i}=\{q\in \mathcal{N}:(A_{m})_{q, i}=0\}, \\ \beta_{j}=\{p\in \mathcal{N}:(A_{m})_{j, p}\neq 0\}, \; \; \; \; \overline{\beta}_{j}=N\setminus \beta_{j}=\{p\in \mathcal{N}:(A_{m})_{j, p}=0\}, \\ \gamma_{j}=\{q\in \mathcal{N}:(A_{m})_{q, j}\neq 0\}, \; \; \; \; \overline{\gamma}_{j}=N\setminus \gamma_{j}=\{q\in \mathcal{N}:(A_{m})_{q, j}=0\}.$

定理2.4  假设$ \beta_i, \; \gamma_i, \; \beta_j, \; \gamma_j $非空, 那么

(ⅰ) 若$ i\in\beta_i $ (即$ i\in\gamma_i $), $ j\in\beta_j $ (即$ j\in\gamma_j $), 且$ 0 $不是$ \mathcal{T}_{i, j, \alpha}(\hat{P}) $的孤立点, 则$ \mathcal{T}_{i, j, \alpha}(P) $无界当且仅当$ 0\in\mathcal{T}_{i, j, \alpha}(A_m) $.

(ⅱ) 若$ i\in\bar{\beta_i} $ (即$ i\in\bar{\gamma_i} $), $ j\in\bar{\beta_j} $ (即$ j\in\bar{\gamma_j} $), 则$ 0\in\mathcal{T}_{i, j, \alpha}(A_m) $且$ \mathcal{T}_{i, j, \alpha}(P) $是无界的.

证  (ⅰ) 先证必要性.

当$ i\in\beta_i $, $ j\in\beta_j $时, 有$ (A_m)_{i, i}\neq0, \; (A_m)_{j, j}\neq0 $. 由原点不是$ \mathcal{T}_{i, j, \alpha}(\hat{P}) $的孤立点, $ \mathcal{T}_{i, j, \alpha}(P) $无界, 令$ \{\mu_l\}_{l\in N} $为$ \mathcal{T}_{i, j, \alpha}(P)\backslash\{0\} $中的序列, $ \mid\mu_l\mid\rightarrow \infty $, 故对任意的$ l\in N $, 有

$ \begin{equation*} \begin{split} \left|\sum\limits^m_{k=0}(A_k)_{i, i}\mu^k_l\right|\left|\sum\limits^m_{k=0}(A_k)_{j, j}\mu^k_l\right|\leq \alpha&\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\mu^k_l\right|\right)\cdot \left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\mu^k_l\right|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\mu^k_l\right|\right)\cdot \left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\mu^k_l\right|\right), \end{split} \end{equation*} $

即

$ \begin{equation*} \begin{split} \left|\sum\limits^m_{k=0}(A_k)_{i, i}\frac{\mu^k_l}{\mu^m_l}\right|\left|\sum\limits^m_{k=0}(A_k)_{j, j}\frac{\mu^k_l}{\mu^m_l}\right|\leq \alpha&\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\frac{\mu^k_l}{\mu^m_l}\right|\right)\cdot\left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\frac{\mu^k_l}{\mu^m_l}\right|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\frac{\mu^k_l}{\mu^m_l}\right|\right)\cdot \left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\frac{\mu^k_l}{\mu^m_l}\right|\right). \end{split} \end{equation*} $

当$ l\rightarrow +\infty $时, 有

$ \begin{equation*} \begin{split} |(A_m)_{i, i}||(A_m)_{j, j}|\leq \alpha&\cdot\left(\sum\limits_{p\in \beta_{i}\backslash\{i\}}|(A_m)_{i, p}|\right)\cdot\left(\sum\limits_{p\in \beta_{j}\backslash\{j\}}|(A_m)_{j, p}|\right)\\ +(1-\alpha)&\cdot\left(\sum\limits_{q\in \gamma_{i}\backslash\{i\}}|(A_m)_{q, i}|\right)\cdot \left(\sum\limits_{q\in \gamma_{j}\backslash\{j\}}|(A_m)_{q, j}|\right), \end{split} \end{equation*} $

故$ 0\in\mathcal{T}_{i, j, \alpha}(A_m) $.

再证充分性.

由于$ 0\in\mathcal{T}_{i, j, \alpha}(A_m) $, 即

$ |(A_m)_{i, i}||(A_m)_{j, j}|\leq \alpha r_i(A_m)r_j(A_m)+(1-\alpha)c_i(A_m)c_j(A_m), $

故$ 0\in\mathcal{T}_{i, j, \alpha}(\hat{P}), $由$ (1.6) $式得$ \infty\in\mathcal{T}_{i, j, \alpha}(P) $ (根据假设$ 0 $不是$ \mathcal{T}_{i, j, \alpha}(\hat{P}) $的孤立点, 从而$ \infty $不是$ \mathcal{T}_{i, j, \alpha}(P) $孤立点), 因此$ \mathcal{T}_{i, j, \alpha}(P) $是无界的.

(ⅱ) 因为$ i\in\bar{\beta_i} $, $ j\in\bar{\beta_j} $, 所以有$ (A_m)_{i, i}=(A_m)_{j, j}=0. $因此$ 0\in\mathcal{T}_{i, j, \alpha}(A_m), $且

$ \begin{equation*} \begin{split} \mathcal{T}_{i, j, \alpha}(P)\backslash\{0\}=&\Bigg\{\mu\in \mathbb{C}\backslash\{0\}:\left|\sum\limits^{m-1}_{k=0}(A_k)_{i, i}\mu^k\right|\left|\sum\limits^{m-1}_{k=0}(A_k)_{j, j}\mu^k\right|\leq\\ &\alpha\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\mu^k\right|\right)\cdot\left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\mu^k\right|\right)\\ &+(1-\alpha)\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\mu^k\right|\right)\cdot\left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\mu^k\right|\right)\Bigg\}\\ =&\Bigg \{\mu\in \mathbb{C}\backslash\{0\}:\left|\sum\limits^{m-1}_{k=0}(A_k)_{i, i}\frac{\mu^k}{\mu^m}\right|\left|\sum\limits^{m-1}_{k=0}(A_k)_{j, j}\frac{\mu^k}{\mu^m}\right|\leq\\ &\alpha\cdot\left(\sum\limits_{p\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{i, p}\frac{\mu^k}{\mu^m}\right|\right)\cdot\left(\sum\limits_{p\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{j, p}\frac{\mu^k}{\mu^m}\right|\right)\\ &+(1-\alpha)\cdot\left(\sum\limits_{q\in N\backslash\{i\}}\left|\sum\limits^m_{k=0}(A_k)_{q, i}\frac{\mu^k}{\mu^m}\right|\right)\cdot\left(\sum\limits_{q\in N\backslash\{j\}}\left|\sum\limits^m_{k=0}(A_k)_{q, j}\frac{\mu^k}{\mu^m}\right|\right)\Bigg \}, \end{split} \end{equation*} $

其中系数矩阵$ (A_m)_{i, p}, \; p\in\; N\backslash\{i\} $和$ (A_m)_{j, p}, \; p\in N\backslash\{j\} $分别至少存在一项同时不为零, 且$ (A_m)_{q, i}, \; q\in N\backslash\{i\} $和$ (A_m)_{q, j}, \; q\in N\backslash\{j\} $分别至少存在一项同时不为零. 因此对于足够大的$ |\mu| $, 有$ \mu\in\mathcal{T}_{i, j, \alpha}(P) $, 即存在实数$ M>0 $, 使得对任意$ |\mu|\geq M\; (\mu\in\mathbb{C}), $有$ \mu\in\mathcal{T}_{i, j, \alpha}(P), $即$ \{\mu\in\mathbb{C}:\mid\mu\mid\geq M\}\subseteq\mathcal{T}_{i, j, \alpha}(P), $故$ \mathcal{T}_{i, j, \alpha}(P) $是无界的.

下面通过算例来比较文中的广义Brauer集与文献[8]中的Brauer集.

例3.1  设矩阵多项式

$ \begin{equation*} \label{eq:7} P_{1}(\lambda)= \left( \begin{array}{cccc} 4.2\lambda^{2}-i & 4\lambda^{2} &0 \\ \lambda-3 & \lambda^{2}+4 &0 \\ -2\lambda+i & \lambda^{2}+2 &2\lambda^{2}-1 \end{array} \right), \end{equation*} $

经计算可得矩阵多项式$ P_{1}(\lambda) $的特征值, 从下面的表 1可见.

矩阵多项式$ P_{1}(\lambda) $的Brauer集和广义Brauer集分别见图 1和图 2.

例3.2  设矩阵多项式

$ \begin{equation*} \label{eq:8} P_{2}(\lambda)=\left( \begin{array}{cccc} 8i\lambda^{2}-2i\lambda+2 & 2i\lambda^{2}+i\lambda+(1+2i)&(-1+i)\lambda^{2}+\lambda+2 \\ -3i\lambda^{2}+5i\lambda+(1+i)&-8i\lambda^{2}+3i\lambda+4i &(2-2i)\lambda^{2}-4\lambda-5i \\ (0.8-i)\lambda^{2}+i\lambda+(1-i) & 0.6i\lambda^{2}-i\lambda &(6-2i)\lambda^{2}+2i \end{array} \right), \end{equation*} $

经计算可得矩阵多项式$ P_{2}(\lambda) $的特征值, 从下面的表 2可见.

矩阵多项式$ P_{2}(\lambda) $的Brauer集和广义Brauer集分别见图 3和图 4.

例3.3  设矩阵多项式

$ \begin{equation*} \label{eq:9} P_{3}(\lambda)=\left( \begin{array}{cccc} 8i\lambda^{2}-2i\lambda+2 & 2i\lambda^{2}+i\lambda+(1+2i)&(-1+i)\lambda^{2}+\lambda+2 \\ -3i\lambda^{2}+5i\lambda+(1+i)&-6i\lambda^{2}+3i\lambda+4i &(2-2i)\lambda^{2}-4\lambda-5i \\ (0.8-i)\lambda^{2}+i\lambda+(1-i) & 6i\lambda^{2}-i\lambda &(6-2i)\lambda^{2}+2i \end{array} \right), \end{equation*} $

经计算可得矩阵多项式$ P_{3}(\lambda) $的特征值, 从下面的表 3可见.

矩阵多项式$ P_{3}(\lambda) $的Brauer集和广义Brauer集分别见图 5和图 6.

注  以上三个例子中, 图中红色点为相应矩阵多项式的特征值, 蓝色区域为特征值的估计区域. 显然例3.1和例3.2中的估计区域均为有界区域, 而例3.3中的估计区域均为无界区域. 无论特征值估计区域是否有界, 广义Brauer集给出的估计区域更为精确.