A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

Front. Math. China 2015, 10(5): 1123–1146
DOI 10.1007/s11464-015-0412-z

Yiyong LI, Qingzhi YANG, Yuning YANG
Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract We give a new definition of geometric multiplicity of eigenvalues of
tensors, and based on this, we study the geometric and algebraic multiplicity
of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with
modulus ρ(A ) have the same geometric multiplicity. We also prove that twodimensional
nonnegative tensors’ geometric multiplicity of eigenvalues is equal
to algebraic multiplicity of eigenvalues.
Keywords Irreducible tensor, Perron-Frobenius theorem, geometrically simple
MSC 15A18, 15A69, 74B99
1 Introduction
Eigenvalue problems of nonnegative tensors, especially, the largest eigenvalue
of a nonnegative tensor, have attracted special attention in the recent years, see
[4,5,7,9,12,14,15]. In particular, the Perron-Frobenius theorem for nonnegative
tensors is related to measuring higher order connectivity in linked objects and
hypergraphs [1,2]. From the matrix theory, we know that the largest eigenvalue
of a nonnegative irreducible matrix is geometrically simple and the
eigenvalues with modulus ρ(A) are equally distributed on the spectral circle.
But the geometric and algebraic multiplicity for a higher order tensor becomes
complicated.
In the literature, the geometric simplicity of the largest eigenvalue (here
we call it the spectral radius) of a nonnegative irreducible tensor was studied
by Chang et al. [4], where some conditions were proposed to ensure the real
geometric simplicity of the spectral radius. Pearson [13] defined essentially
positive tensors and proved that the spectral radius of an essentially positive
Received April 4, 2014; accepted July 4, 2014
Corresponding author: Qingzhi YANG, E-mail: qz-yang@nankai.edu.cn