复Hadamard矩阵的等价性
Equivalence of Complex Hadamard Matrices
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中文总结 AI 辅助
本文研究复Hadamard矩阵的等价性，提出确定矩阵等价性及自同构群的算法，并应用于文献中大量矩阵以确定其自同构群。
中文摘要 AI 辅助
在等价性或同构的语境下，对称性是任何离散结构研究中的一个基本且自然的概念。对称性对于非离散结构也很重要，但其处理可能更具挑战性，因此或许常常被忽视。这在许多关于复Hadamard矩阵的研究中也是如此，复Hadamard矩阵是指具有单模复元素且满足方程$HH^{\dagger} = nI$的矩阵，其中$H^{\dagger}$是$H$的共轭转置。在当前工作中，考虑了复Hadamard矩阵的等价性，并提出了用于确定矩阵等价性和矩阵自同构群的算法。这些算法被用于确定文献中大量复Hadamard矩阵的自同构群。
英文摘要
Symmetry in the context of equivalence or isomorphism is a fundamental and natural concept in any study of discrete structures. Symmetries are also important for non-discrete structures, but their treatment can be more challenging and is perhaps therefore often overlooked. This holds for many studies of complex Hadamard matrices, that is, matrices with unimodular complex entries satisfying the equation $HH^{\dagger} = nI$, where $H^{\dagger}$ is the conjugate transpose of $H$. In the current work, equivalence of complex Hadamard matrices is considered, and algorithms for determining equivalence of matrices and the automorphism group of a matrix are presented. The algorithms are used to establish the automorphism group of a large number of complex Hadamard matrices from the literature.
发表机构
- Aalto University School of Electrical Engineering（阿尔托大学电气工程学院）
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