The tensor eigenvalue methods for the positive definiteness identification problem

The main purposes of this thesis are to solve the positive definiteness identification problems by using some tensor eigenvalue methods. The positive definiteness identification problems arise in numerous fields of mathematics and its applications. Recently, Qi presented the concepts of H-eigenvalues/Z-eigenvalues of a real n-dimensional supersymmetric tensor, and explored their practical applications in determining positive definiteness of an even degree multivariate form. In the thesis, we proposed tensor eigenvalue methods for the positive definiteness identification problems. Using formulas of the resultants of polynomial equations systems, we present two viable algorithms to calculate the smallest H-eigenvalue/Z-eigenvalue for a real supersymmetric tensor respectively. Numerical results show that our algorithms are feasible and efficient for identifying the positive definiteness of quartic forms with three variables.;The main contributions of this thesis are as follows.;In the thesis, we first present a complete explicit criterion for the positive definiteness of a general quartic form of two variables.;Secondly, we present an H-eigenvalue method for the positive definiteness identification problem. At first we apply the D'Andrea-Dickenstein version of the classical Macaulay formulas of resultants to compute the symmetric hyperdeterminant of an even order supersymmetric tensor. By using the supersymmetry property, we give detailed computation procedures for the Bezoutians and specified ordering of monomials in this approach. We then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an H-eigenvalue method for the positive definiteness identification problem of a quartic form of three variables.;Thirdly, for an even order tensor, we first establish a formula of the E-characteristic polynomial of its tensor based on the Macaulay's formulas of resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this bound is tough when m and n is small.;Finally, we present a Z-eigenvalue method to identify the positive definiteness for quartic forms of three variables. We first construct a limiting E-characteristic polynomial for an even order supersymmetric tensor in the irregular case. Then we apply the D'Andrea-Dickenstein version of the classical Macaulay formula of a resultant to establish a formula of the (limiting) E-characteristic polynomial for a regular (irregular) fourth order three dimensional real supersymmetric tensor. Using such a formula, we propose an algorithm which is efficient on determining positive definiteness of a quartic form of three variables. Some numerical results of the H-eigenvalue method and the Z-eigenvalue for identifying positive definiteness of quartic forms with three variables are reported.;Keywords. Supersymmetric tensor, symmetric hyperdeterminant, eigenvalue method, characteristic polynomial, positive definiteness.