Hermitian unitary matrices with modular permutation symmetry
Hermitian unitary matrices with modular permutation symmetry
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具有模置换对称性的埃尔米特酉矩阵
DOI：
10.1016/j.laa.2014.12.011
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发表时间：
2015
影响因子：
1.1
通讯作者：
Taksu Cheon
中科院分区：
文献类型：
--
作者：
Ondrej Turek;Taksu Cheon
本文研究了Hermitian酉矩阵S∈ Cn，n，其中存在r≥ 0和t> 0，使得S的元素满足|S J J| r，且|S·J·K| = t对于所有j，k= 1，...，n，j <$k。我们得出必要条件的比率d：= r/t，并表明，这些条件是非常严格的情况下，除了当n是偶数和S的对角线元素的总和为零。对于属于某些区间的d，构造矩阵族S的例子。真实的矩阵S的情况下，检查更详细。证明了一个真实的S只能存在于d= n2 − 1，或n为偶数且n2 + d1（mod 2）。我们给出了d≥ n 4− 3 2的真实的S的结构的详细描述，并根据某些对称（v，k，λ）-设计的存在性得到了它存在的一个充分必要条件.我们证明了不存在真实的S，其中d∈（n 6− 1，n 4− 3 2）.给出了厄米酉矩阵的参数化，并将其推广到一般酉矩阵。文章最后简要说明了所研究的矩阵在图上量子力学中的作用。
We study Hermitian unitary matrices S∈ C n, n with the following property: There exist r≥ 0 and t> 0 such that the entries of S satisfy| S j j|= r and| S j k|= t for all j, k= 1,…, n, j≠ k. We derive necessary conditions on the ratio d:= r/t and show that these conditions are very restrictive except for the case when n is even and the sum of the diagonal elements of S is zero. Examples of families of matrices S are constructed for d belonging to certain intervals. The case of real matrices S is examined in more detail. It is demonstrated that a real S can exist only for d= n 2− 1, or for n even and n 2+ d≡ 1 (mod 2). We provide a detailed description of the structure of real S with d≥ n 4− 3 2, and derive a sufficient and necessary condition of its existence in terms of the existence of certain symmetric (v, k, λ)-designs. We prove that there exists no real S with d∈(n 6− 1, n 4− 3 2). A parametrization of Hermitian unitary matrices is also proposed, and its generalization to general unitary matrices is given. At the end of the paper, the role of the studied matrices in quantum mechanics on graphs is briefly explained.