Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4599265 | Linear Algebra and its Applications | 2015 | 25 Pages |
Abstract
We study Hermitian unitary matrices SâCn,n with the following property: There exist râ¥0 and t>0 such that the entries of S satisfy |Sjj|=r and |Sjk|=t for all j,k=1,â¦,n, jâ 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=n2â1, or for n even and n2+dâ¡1(mod2). We provide a detailed description of the structure of real S with dâ¥n4â32, 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 dâ(n6â1,n4â32). 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
OndÅej Turek, Taksu Cheon,