Author:  Wang, Fei 
Title:  The tensor eigenvalue methods for the positive definiteness identification problem 
Degree:  Ph.D. 
Year:  2006 
Subject:  Hong Kong Polytechnic University  Dissertations Eigenvalues Calculus of tensors Mathematical analysis 
Department:  Dept. of Applied Mathematics 
Pages:  vii, 110 leaves ; 30 cm 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2069732 
URI:  http://theses.lib.polyu.edu.hk/handle/200/687 
Abstract:  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. Effective methods intended to obtain a reliable answer to the problems are of doubtless theoretical and practical interest. Recently, to do this, Qi presented the concepts of Heigenvalues/Zeigenvalues of a real ndimensional supersymmetric tensor, and proved that Heigenvalues/Zeigenvalues exist for an even order real supersymmetric tensor A, and A is positive definite if and only if all of its Heigenvalues/Zeigenvalues are positive. Based on this, 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 Heigenvalue/Zeigenvalue 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 Heigenvalue method for the positive definiteness identification problem. At first we apply the D'AndreaDickenstein 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 Heigenvalue 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 Echaracteristic polynomial of its tensor based on the Macaulay's formulas of resultants, then give an upper bound for the degree of that Echaracteristic polynomial. Examples illustrate that this bound is tough when m and n is small. Finally, we present a Zeigenvalue method to identify the positive definiteness for quartic forms of three variables. We first construct a limiting Echaracteristic polynomial for an even order supersymmetric tensor in the irregular case. Then we apply the D'AndreaDickenstein version of the classical Macaulay formula of a resultant to establish a formula of the (limiting) Echaracteristic 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 Heigenvalue method and the Zeigenvalue for identifying positive definiteness of quartic forms with three variables are reported. 
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