On the Pattern of a Unitary Matrix
نویسنده
چکیده
Given a digraph D on n vertices, a matrix A of size n is said to have pattern D, if it has entry Aij 6= 0 if and only if (vi, vj) is an arc of D. We give a necessary condition, which is called strong quadrangularity, for a digraph to be the pattern of a unitary matrix. With the use of such a condition, we show that a line digraph L (D) is the pattern of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and L (D) is the pattern of a unitary matrix then L (D) is Hamiltonian. Strong quadrangularity is also used to prove that n-paths, n-paths with loops at each vertex, n-cycles, directed trees and trees are not patterns of unitary matrices.
منابع مشابه
New Bases for Polynomial-Based Spaces
Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...
متن کاملON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS
Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...
متن کاملSome bounds on unitary duals of classical groups - non-archimeden case
We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$-adic groups. Roughly, they can show up only if the central character of the inducing irreducible cuspidal representation is dominated by the square root of the modular character of the minimal parabolic subgroup. For unitarizable subquotients...
متن کاملProperties of matrices with numerical ranges in a sector
Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...
متن کاملOn local gamma factors for orthogonal groups and unitary groups
In this paper, we find a relation between the proportionality factors which arise from the functional equations of two families of local Rankin-Selberg convolutions for irreducible admissible representations of orthogonal groups, or unitary groups. One family is that of local integrals of the doubling method, and the other family is that of local integrals expressed in terms of sph...
متن کامل