Mathematics

Eigenvalues of tensors are a direct generalization of the concept of eigenvalues of matrices to arrays of higher order and, as the latter, have potential for a widespread amount of applications. For instance, symmetric tensors can be seen as multivariate homogeneous polynomials and the largest real eigenvalue of a symmetric tensor then yields the maximum of the corresponding polynomial on the unit sphere. Unlike for symmetric matrices, the number of real eigenvalues of a real tensor is not generically constant. In this talk I will present recent results on the expected number of real eigenvalues of a symmetric tensor, whose entries are Gaussian random variables. To be more precise, this model of a random symmetric tensors is the generalization of the Gaussian Orthogonal Ensemble (GOE) to higher order. The computation of the expected absolute determinant of a GOE matrix will play a role.