Further results on eigenvalues of symmetric decomposable tensors from multilinear dynamical systems

Eigenvalues of tensors play a crucial role in many practical problems. In this paper, we present several new properties on eigenvalues of symmetric decomposable tensors from multilinear dynamical systems. Under orthonormal conditions, we provide a new proof for a conjecture such that the spectral radius of an orthogonal decomposable tensor is included in its coefficients. Moreover, if there is a generating vector orthogonal to other generating vectors for a symmetric decomposable tensor, it is proved that the vector is an eigenvector of the tensor. As an application of the conjecture, a sufficient region is given to guarantee the asymptotically stability of the multilinear system and numerical example verifies its performance.