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yiju66@hotmail.com (Y. Wang).

1The work of this author was supported by the Hong Kong Research Grant Council and the National Natural Science

Foundation of China (No. 60572135).

2The work of this author was supported by the Research Grant Council of Hong Kong.

3The work of this author was supported by a Hong Kong Polytechnic University Postdoctoral Fellowship.

J. Math. Anal. Appl. 329 (2007) 1218–1229

www.elsevier.com/locate/jmaa

The degree of the E-characteristic polynomial

of an even order tensor

Guyan Nia,1, Liqun Qib,∗,2, Fei Wangc, Yiju Wangc,d,3

aDepartment of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, PR China

bDepartment of Mathematics, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

cDepartment of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

dSchool of Operations Research and Management Sciences, Qufu Normal University, Rizhao,

Shandong 276800, PR China

Received 15 March 2006

Available online 17 August 2006

Submitted by Jerzy Filar

Abstract

The E-characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining

the positive definiteness of an even degree multivariate form. In this paper, for an even order tensor, we

first establish the formula of its E-characteristic polynomial by using the classical Macaulay formula of

resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate

that this bound is attainable in some low order and dimensional cases.

© 2006 Elsevier Inc. All rights reserved.

Keywords: Tensor; E-Characteristic polynomial; E-Eigenvalue; Resultant; Upper bound

*Corresponding author.

E-mail addresses: guyan-ni@163.com (G. Ni), maqilq@cityu.edu.hk (L. Qi), fei.wang@polyu.edu.hk (F. Wang),

0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2006.07.064

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For example, if m = 4 and n = 3, then d = 27.

On the other hand, the degree of the E-characteristic polynomial is lower than this. In [26],

E-eigenvalues and E-characteristic polynomials were further discussed. The definitions of eigen-

values, eigenvectors, E-eigenvalues, E-eigenvectors, E-characteristic polynomials were gener-

alized to nonsymmetric tensors. It was shown in [26] that the degree of the E-characteristic

polynomial of an mth order n-dimensional tensor varies for different tensors. Sometimes there

G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229

1219

1. Introduction

An mth order tensor is an m-way array whose entries accessed via m indices. It arises in di-

verse fields such as signal and image processing, data analysis, nonlinear continuum mechanics,

higher-order statistics, as well as independent component analysis [5,7,8,12,15,16,19,22,29,31].

It is well known that supersymmetric tensors and homogeneous polynomials are bijectively as-

sociated [8,13], and when m is even, the positive definiteness of a homogeneous polynomial

plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s

direct method in automatic control [1–4,9,11,14,30]. Motivated by this issue, Qi [24] introduced

the concepts of eigenvalues and E-eigenvalues of a supersymmetric tensor, and established their

close relationship with the theory of resultants [6,10,28].

An mth degree homogeneous polynomial form of n variables f(x) can be represented as the

product of two tensors

f(x) ≡ Axm=

n

?

i1,...,im=1

ai1···imxi1···xim,

(1.1)

where tensor A is a supersymmetric tensor, i.e., its entries ai1···imare invariant under any per-

mutation of their indices i1,...,im= 1,...,n, and xmis a supersymmetric tensor with entries

xi1xi2···xim.

A supersymmetric tensor A is called positive definite if it satisfies

Axm> 0,

∀x ∈ Rn, x ?= 0.

For a vector x ∈ Cn, we denote its ith component by xi. By the tensor product [27], Axm−1

is a vector in Cnwhose ith component is

n

?

i2,...,im=1

In [24], Qi introduced eigenvalues, eigenvectors, E-eigenvalues, E-eigenvectors, characteristic

polynomials and E-characteristic polynomials for supersymmetric tensors. When m ? 3, eigen-

values and E-eigenvalues may not be real. An eigenvalue (E-eigenvalue) with a real eigenvector

(E-eigenvector) is called an H-eigenvalue (Z-eigenvalue). An even order supersymmetric tensor

always has H-eigenvalues and Z-eigenvalues. It is positive (semi)definite if and only if all of

its H-eigenvalues or all of its Z-eigenvalues are positive (nonnegative). A complex number is an

eigenvalue of a supersymmetric tensor if and only if it is a root of the characteristic polynomial of

that tensor. Based upon these, an H-eigenvalue method for the positive definiteness identification

problem was developed in [21].

By [24], the degree of the characteristic polynomial of an mth order n-dimensional supersym-

metric tensor is

d = n(m−1)n−1.

aii2···imxi2···xim.

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solutions of the following polynomial equation system:

?Axm−1= λx,

If x is real, then λ is also real. In this case, λ and x are called a Z-eigenvalue of A and

a Z-eigenvector of A associated with the Z-eigenvalue λ, respectively.

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G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229

may be zero E-characteristic polynomials. The maximum of degrees of E-characteristic polyno-

mial of mth order n-dimensional tensors is denoted as d(m,n) when m is even. When m is odd,

the E-characteristic polynomial of an mth order n-dimensional tensor only contains even degree

terms. Thus, the maximum of degrees of E-characteristic polynomial of mth order n-dimensional

tensors is denoted as 2d(m,n) when m is odd. It was shown in [26], that d(1,n) ≡ 1, d(2,n) = n,

d(m,2) = m for m ? 3 and

d(m,n) ? mn−1+mn−2+···+m

for m,n ? 3. When m = 4 and n = 3, (1.2) gives an upper bound 20 for d(m,n). This shows that

the degree of the E-characteristic polynomial is much lower than the degree of the characteristic

polynomial, and a Z-eigenvalue method for the positive definiteness identification problem may

be better than the H-eigenvalue method.

The upper bound for d(m,n) given in (1.2) can be improved. In this paper, we do this when m

is even. In particular, we show that d(4,3) = 13, which is much smaller than 20, the upper bound

given in (1.2) and 27, the degree of the characteristic polynomial when m = 4 and n = 3. In [20],

using the result d(4,3) = 13 in this paper, a Z-eigenvalue method for the positive definiteness

identification problem for a quartic form of three variables is developed. Numerical results show

that this method is better than the existing global polynomial optimization methods [23], applied

to this problem.

In the following sections, for an even order tensor, we first establish the formula of its E-

characteristic polynomial by using the classical Macaulay formula of resultants, then give an

upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this

bound is tough in some low order and dimensional cases.

In [25], geometric meanings of Z-eigenvalues are discussed. In [26], it was also shown that

E-eigenvalues are invariant under co-ordinate changes in the sense of tensor analysis used in

nonlinear mechanics [12,29]. This shows an additional merit of E-eigenvalues. Independently,

with a variational approach, Lim also defines eigenvalues of tensors in [17] in the real field.

The l2eigenvalues of tensors defined in [17] are Z-eigenvalues in [24], while the lkeigenvalues

of tensors defined in [17] are H-eigenvalues in [24]. Notably, Lim [17] proposed a multilin-

ear generalization of the Perron–Frobenius theorem based upon the notion of lkeigenvalues

(H-eigenvalues) of tensors.

(1.2)

2. A formula of the E-characteristic polynomial

In this section, we will first review the definition of E-eigenvalues, E-characteristic polyno-

mials, and their properties. Then we will review the classical Macaulay formula of the resultant

for a polynomial system, stated in [6]. Finally, we will use the Macaulay formula to establish a

formula of the E-characteristic polynomial of an even order tensor A.

Definition 2.1. For a real tensor A, a number λ ∈ C is called an E-eigenvalue of A and a nonzero

vector x ∈ Cnis called an E-eigenvector of A associated with the E-eigenvalue λ, if they are

xTx = 1.

(2.3)

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Sn=?xα: |α| =¯d, xd1

Consider the system of homogeneous equations of degree¯d:

⎧

⎪⎩

G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229

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It was shown in [24] that Z-eigenvalues always exist for a real supersymmetric tensor A,

and when the order of A is even, A is positive definite if and only if all of its Z-eigenvalues

are positive. Thus, the smallest Z-eigenvalue of an even order supersymmetric tensor A is an

indicator of the positive definiteness of A.

Assume that m is even. Let A be an mth order tensor and

Fλ(x) = Axm−1−λI(x)x = 0,

where I(x) = (xTx)

characteristic polynomial φ(λ) of A, i.e.,

φ(λ) = Res?Fλ(x)?.

The tensor A is called regular if there is no vector x ?= 0 such that

?Axm−1= 0,

The following theorem was shown in [26].

(2.4)

m−2

2 . Then the resultant of Fλ(x), denoted by Res(Fλ(x)), is the E-

xTx = 0.

Theorem 2.1. Assume that m,n ? 2. Let d(m,n) be the maximum of degrees of E-characteristic

polynomials of mth order n-dimensional tensors. Then the following statements hold:

(a) An E-eigenvalue of A is a root of the E-characteristic polynomial φ. If A is regular, then a

complex number is an E-eigenvalue of A if and only if it is a root of φ.

(b) d(2,n) = n. For m ? 3, d(m,2) = m. For m,n ? 3,

d(m,n) ? mn−1+···+m.

This theorem holds for all m,n ? 2. But in this paper, we only discuss the case that m is even.

We denote by k[x1,...,xn] the collection of all polynomials in x1,...,xnwith coefficients

in k, where k is a field.For homogeneouspolynomials F1,F2,...,Fn∈ C[x1,x2,...,xn] of total

degrees d1,d2,...,dn, set

¯d =

n

?

i=1

(di−1)+1 =

n

?

i=1

di−n+1.

Let S be the set of the monomials xα= xα1

following n sets:

S1=?xα: |α| =¯d, xd1

...

1···xαn

n of total degree¯d and divide it into the

1divides xα?,

S2=?xα: |α| =¯d, xd1

1does not divide xαbut xd2

2does?,

1,...,xdn−1

n−1do not divide xαbut xdn

ndoes?.

(2.5)

⎪⎨

xα/xd1

···

xα/xdn

1·F1(x) = 0for all xα∈ S1,

n ·Fn(x) = 0for all xα∈ Sn.

(2.6)