| 摘要: |
| 本文通过引入酉矩阵${U}$研究了矩阵多项式$P(\lambda)$的数值域与其子块数值域之间的关系, 并给出了相关的理论分析. 与以往研究数值域和子块数值域包含关系(即$W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(P)_{{I}_{l}}\subset W(P)$)不同, 本文通过酉相似变换, 在一定条件下建立了以下双向包含关系: 对于所有酉矩阵${U}$, $UPU^*$的子块数值域的并集等于$P(\lambda)$的数值域, 即$$ \bigcup\limits_{\substack{U\in \mathbb
{U}(\mathcal{H})}} W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(UPU^*)_{{I}_{l}}= W(P),$$其中$\mathbb{U}(\mathcal{H})$表示空间$\mathcal{H}=\mathbb{C}^{n}$上的所有酉矩阵构成的集合, $W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(UPU^*)_{{I}_{l}}$表示$UPU^*$的子块数值域. |
| 关键词: 矩阵多项式 数值域 子块数值域 |
| DOI： |
| 分类号:O177.1 |
| 基金项目:国家自然科学基金项目;内蒙古自然科学基金项目 |
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| ON THE RELATION BETWEEN THE NUMERICAL RANGE OF A MATRIX POLYNOMIAL AND ITS BLOCK SUBMATRIX NUMERICAL RANGES |
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chen shu li, qi ya ru
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Inner Mongolia University of Technology
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| Abstract: |
| This paper investigates the relationship between the numerical range of a matrix polynomial $P(\lambda)$ and the numerical ranges of its block submatrices through unitary matrices $U$, with complete theoretical analysis. Unlike previous studies focusing on the containment relation between the numerical range and the block submatrices numerical ranges (i.e., $ W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(P)_{{I}_{l}}\subset W(P)$), Under specified conditions, this paper establishes the following mutual inclusion relation via unitary similarity transformations: the union of the block submatrices numerical ranges of $UPU^*$ over all unitary matrices $U$ equals the numerical range of $P(\lambda)$, i.e., $$ \bigcup\limits_{\substack{U\in \mathbb
{U}(\mathcal{H})}} W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(UPU^*)_{{I}_{l}}= W(P),$$ where $\mathbb{U}(\mathcal{H})$ denotes the set of all unitary matrices acting on the space $\mathcal{H}=\mathbb{C}^{n}$, and $W_{\mathcal{H}_{i_{1}}×\cdots×\mathcal{H}_{i_{l}}}(UPU^*)_{{I}_{l}}$ represents the numerical range of the block submatrices of $UPU^*$. |
| Key words: Matrix polynomial Numerical range Block submatrix numerical range |
