Biquadratic Tensors: Eigenvalues and Structured Tensors
Liqun Qi, Chunfeng Cui
TL;DR
This work extends M-eigenvalues to nonsymmetric biquadratic tensors and proves that a biquadratic tensor is positive semidefinite (definite) if and only if all its M-eigenvalues are nonnegative (positive). It introduces a Gershgorin-type theorem for M-eigenvalues and analyzes diagonally dominated cases, establishing PSD/PD results for several structured tensor classes (including Z-, M-, strong M-, B0-, and B-biquadratic tensors). A Riemannian LBFGS algorithm is developed to compute the smallest M-eigenvalue on the product of two unit spheres, with convergence guarantees and practical performance demonstrated through numerical experiments on elasticity, covariance, and related problems. The results provide a practical toolkit for assessing strong ellipticity and positivity properties in diverse applications, and open avenues for tighter bounds and SOS decompositions in biquadratic tensor theory.
Abstract
The covariance tensors in statistics{, elasticity tensor in solid mechanics, Riemann curvature tensor in relativity theory are all biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this paper, we consider nonsymmetric biquadratic tensors, and study possible conditions and algorithms for identifying positive semi-definiteness and definiteness of such biquadratic tensors. We extend M-eigenvalues to nonsymmetric biquadratic tensors, prove that a general biquadratic tensor has at least one M-eigenvalue, and show that a general biquadratic tensor is positive semi-definite if and only if all of its M-eigenvalues are nonnegative, and a general biquadratic tensor is positive definite if and only if all of its M-eigenvalues are positive. We present a Gershgorin-type theorem for biquadratic tensors, and show that (strictly) diagonally dominated biquadratic tensors are positive semi-definite (definite). We introduce Z-biquadratic tensors, M-biquadratic tensors, strong M-biquadratic tensors, B$_0$-biquadratic tensors and B-biquadratic tensors. We show that M-biquadratic tensors and symmetric B$_0$-biquadratic tensors are positive semi-definite, and that strong M-biquadratic tensors and symmetric B-biquadratic tensors are positive definite. A Riemannian LBFGS method for computing the smallest M-eigenvalue of a general biquadratic tensor is presented. Numerical results are reported.