Abstract
Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for a circulant tensor to be positive semi-definite.
Highlights
As a natural extension of circulant matrices, circulant tensors naturally arise from stochastic process and spectral hypergraph theory
We study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process
We show that when the associated tensor is a nonnegative tensor, λ0(A) is the largest H-eigenvalue of A. This confirms the results in circulant hypergraphs and directed circulant hypergraphs
Summary
In the three sections, we give various conditions for an even order circulant tensor to be positive semi-definite. The Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. We present some necessary conditions, sufficient conditions, and necessary and sufficient conditions for an even order circulant tensor with a diagonal root tensor to be positive semi-definite. An algorithm for determining positive semi-definiteness of an even order circulant tensor with a diagonal root tensor, and its numerical experiments are presented. We study what are the concerns on the properties of circulant tensors in these applications
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