On the digraph of a unitary matrix

Details On the digraph of a unitary matrix On the digraph of a unitary matrix

2002-05-16

arxiv.org/abs/math/0205187v2

SIAM Journal on Matrix Analysis and Applications, Volume 25, Number 1, pp. 295-300, July 2003

Mathematics

Combinatorics

6 pages

Scientific paper

10.1137/S0895479802410293

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.

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