Applied Mathematics Seminar

Eigenvalues of Tensors and Their Applications

*Speaker: Professor Liqun Qi
Chair Professor of Applied Mathematics
Department of Applied Mathematics
The Hong Kong Polytechnic University
*Date: Thursday April 28th
*Time: 11am
*Room: RC-4082

In this talk, we extend eigenvalues of real symmetric matrices to real supersymmetric tensors. The eigenvalues are the roots of the
characteristic polynomial, which is a one-dimensional polynomial associated with the hyperdeterminant of that tensor, where the concept of hyperdeterminants was introduced by Cayley in 1845. The product of all the
eigenvalues is equal to the value of the hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that tensor, multiplied with a factor depending upon the order and dimension of
that tensor. We also define the rank for a supersymmetric tensor. The rank of an $m$th order $n$-dimensional supersymmetric tensor is an
integer $r$ satisfies $0 \le r \le n$. Under an orthogonal transformation, the homogeneous polynomial defined by the tensor can be converted to a homogeneous polynomial of $r$ variables.

Currently, the new theory has two applications. One is for determining the positive definiteness of a multivariate polynomial with small degree and dimension. This topic is important in stability analysis of automatic control. Another application is for the classification of algebraic curves and surfaces, a research topic started by Issac Newton and may have implications in higher-order partial differential equations. More applications are expected.