Research Concerning The Eigenvalue Of Third-Order Symmetric Tensors

In the nineteenth century, in the research of differential geometry, Gauss, Riemann, Christoffel, etc., introduced the concept of tensors. At the beginning of the twentieth century, Ricci, Levi-Civita, etc., further developed the analysis of tensors and made it a mathematical discipline. In1916, Einstein applied the tensors analysis to the research of general relativity. And this made the analysis of tensors an important tool in the study of continuum mechanics, theoretical physics and many other areas of science and engineering. In2005, Liqun Qi firstly gave the super characteristic polynomial, eigenvalues and E-eigenvalues of a real supersymmetric tensor in [19], and also proved some properties of eigenvalues.In this paper, the author mainly studies the three-order supersymmetric tensors. First of all, we summarize the earlier results, and do some research on the properties of three-order supersymmetric tensors according to the proper-ties of eigenvalues of the second-order tensors. We mainly divide the paper into three chapters.In chapter1, we simply introduce the background of this paper, some basic notations, and also some fundamental concepts and results concerning the eigenvalues of tensors.In chapter2, we mainly study the eigenvalues of three-order supersym-metric tensors and summarize the research of Qi([19],[20]). Furthermore, we give some examples to explain these results and also obtain the following the-orems:Theorem2.1:Assume that A is a three-order n-dimension supersym- metric tensor. Then A always has H-eigenvalues.Theorem2.2: Assume that A is a three-order supersymmetric tensor. If ζ is a eigenvector of A associated with the eigenvalue λ0, then kζ is also a eigenvector of A associated with the eigenvalue λ0for any k≠0. And when n=2, if ζ1,ζ2are two eigenvectors of A associated with the eigenvalue λ0, then if A1122=A111A122, A2122=A211A222,then k1ζ1+k2ζ2are also eigenvectors of A associated with the eigenvalue λ0.Theorem2.3: If λ and μ are two eigenvalues of three-order supersym-metric tensor A,λ≠μ. x and y are their eigenvectors, respectively. Then x and y are linear independent.In chapter3. we study some properties of E-eigenvalues of three-order tensors, which extend some results of Qi[19]. We obtain the following results:Theorem3.1: Assume that A a three-order n-dimension tensor. If λ is an E-eigenvalue of A. Then λ is a root of the E-characteristic polynomial of A. If A is regular, then a complex number is an E-eigenvalue of A if and only if it is a root of the E-characteristic polynomial of A.Theorem3.2: Assume that A a three-order n-dimension tensor. If A is regular, then A has cither only a finite number of E-eigenvalue, or all complex numbers are its E-eigenvalues, i.e., it is singular. When A has only a finite number of E-eigenvalues, A has at most2d(3,n) E-eigenvalues.Theorem3.3: Assume that A, B are three-order n-dimension tensors, If A, B are orthogonal similar, then they have the same E-eigenvalue; If B=P3A, where P is n×n real orthogonal matrix, λ is an E-eigenvalue of A, x is the eigenvector of A associated with λ, then λ is also an E-eigenvalue of B, and y=Px is the eigenvector of B associated with λ. Theorem3.3:Assume that A is a three-order n-dimension supersym-metric tensor. Then A always has E-eigenvalues and E-eigenvectors.