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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
1221
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)
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xα/xm−1
i
·Fi(x) = 0
for i = 1,...,n.
Regarding the monomials of total degree¯d as unknowns. Then we get a system of N linear
equations in N unknowns, where N =?n(m−1)
of Mλobtained by deleting all rows and columns corresponding to reduced monomials xα.
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G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
Since Fi has degree di, it follows that xα/xdi
on the left side of (2.6) can be written as a linear combination of monomials of total degree¯d.
Suppose that there are N such monomials, where N =?¯d+n−1
the monomials of total degree¯d as unknowns, we get a system of N linear equations in N
unknowns.
Denote the coefficient matrix of the N × N system of equations by M. A monomial xαof
total degree¯d is called reduced if xdi
M?the submatrix of the coefficient matrix of (2.6) obtained by deleting all rows and columns
corresponding to reduced monomials xα.
Macaulay [18] gave the following formula for the resultant as a quotient of two determinants.
i· Fi has total degree¯d. Thus each polynomial
?. Then observe that the total
n−1
number of equations is the number of elements in S1∪···∪Sn, which is also N. Thus, regarding
idivides xαfor exactly one i, where i = 1,...,n. Denote
Theorem 2.2. When F1,F2,...,Fnare universal polynomials, the resultant of {F1,F2,...,Fn}
is given by
Res = ±det(M)
det(M?).
(2.7)
Furthermore, if k is a field and F1,F2,...,Fn∈ k[x1,x2,...,xn], then the formula for Res holds
whenever det(M?) ?= 0.
Now we discuss the resultant of Fλ(x) based on the above discussion. For convenience, we
denote (2.4) by
⎛
⎜
Fn(x)
Obviously,
Fλ(x) =
⎜
⎝
F1(x)
F2(x)
...
⎞
⎟
⎟
⎠= 0 and
¯ F(x) = Axm−1.
Fi(x) =¯ Fi(x)−λI(x)xi
Let d1,...,dn= m−1. Then we have
n
?
Let S be the set of the monomials xα= xα1
as (2.5), where d1= d2= ··· = dn= m−1.
Consider the system of homogeneous equations of degree¯d:
for all xα∈ Si,
for i = 1,...,n.
(2.8)
¯d =
i=1
(m−1−1)+1 = n(m−2)+1.
1···xαn
n of total degree¯d and divide it into n sets
(2.9)
n−1
?.
Denote by Mλthe coefficient matrix of the N ×N system of equations and M?
λthe submatrix
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This procedure is equivalent to perform the elementary row operation on the corresponding row
of the matrix Mλ.
Repeat this process till it cannot be executed. Then we get a new system of “linear” equations
with a coefficient matrix, denoted by ˆ M.
Denote by r the number of elements in
xm−2
1
of rows containing λ in ˆ M is r and dM= r.
G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
1223
Theorem 2.3. Assume that A is a universal mth order n-dimensional supersymmetric tensor and
m is even. Then the E-characteristic polynomial of A is given by
φ(λ) = ±det(Mλ)
det(M?
λ).
(2.10)
Furthermore, if A is a real tensor, then the above formula holds whenever det(M?
λ) ?= 0.
Proof. The conclusion follows from Theorems 2.1 and 2.2.
2
3. An upper bound of the degree of φ(λ)
In this section, we consider the degree of the E-characteristic polynomial of A. Assume that
A is a universal even order tensor. We assume that m,n ? 2. We rewrite φ(λ) as follows:
φ(λ) =
d
?
i=0
ciλi,
where d is the degree of φ(λ) with respect to λ, ci’s are homogeneous polynomials in elements
of A.
Denote the degrees of det(Mλ) and det(M?
Obviously, we have
λ) with respect to λ by dMand dM?, respectively.
d(m,n) ? d = dM−dM?.
(3.11)
Theorem 3.1.
dM=
?(n−1)(m−1)+1
n−1
?
.
Proof. Consider the elements of Mλthat contains λ. We rewrite (2.9) as follows:
?xα/xm−1
If there exist α(i)∈ Siand α(j)∈ Sjfor some 1 ? i ?= j ? n such that
xα(i)/xm−2
ij
,
then the equation
?xα(j)/xm−1
can be replaced by
?xα(j)/xm−1
i
?¯ Fi−λ?xα/xm−2
i
?I(x) = 0 for all xα∈ Si, i = 1,...,n.
(3.12)
= xα(j)/xm−2
j
?¯ Fj−λ?xα(j)/xm−2
?¯ Fj−?xα(i)/xm−1
j
?I(x) = 0
?¯ Fi= 0.
ji
S1
∪···∪
Sn
xm−2
n
. It is easy to observe that the number
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xα
xm−2
i
Denote it by x(i,t,α), where t is the first index of α such that αt? 1. It is clear that 1 ? t ? i,
and
x(i,t,α) ∈ˆSt,
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G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
We now compute r. Let S?=?n
S?
i=1S?
i = 1,2,...,n,
i, where
i=?xα: xαxm−2
S??=?xα: |α| = (n−1)(m−2)+1?.
It is clear that
xm−2
i
ously, S?⊆ S??. On the other hand, since S1,...,Snconstitute a partition of the set S, combining
this with the definition of these sets, we conclude that for any xα∈ S??, there exists at least one
index i such that 1 ? i ? n and xαxm−2
i
theory, we can compute the cardinality of the set S??equals
?(n−1)+(n−1)(m−2)+1
This completes the proof.
2
Now, we consider the degree of λ in det(M?
set B.
i
∈ Si
?,
and
Si
= S?
ifor all i = 1,...,n. Moreover, we claim that S?= S??. In fact, obvi-
∈ Si, so xα∈ S?. Hence, S?= S??. From the combinatory
?(n−1)(m−1)+1
n−1
?
=
n−1
?
.
λ). Denote by |B| the number of entries of the
Theorem 3.2. dM? = |ˆS??|, whereˆS??is defined by (3.15).
Proof. By the definition of reduced monomials, we know that a monomial xαof total degree¯d
is not reduced if and only if there exist at least two distinct indices i,j ? 1 such that αi? m−1
and αj? m − 1. So all entries of Snare reduced, and all nonreduced monomials can be divided
into n−1 sets according to (2.5) as follows:
ˆS1=?xα: xα∈ S1, αi? m−1 for some 2 ? i ? n?,
...
ˆSn−1=?xα: xα∈ Sn−1, αn? m−1?.
Recall that the matrix M?
rows and columns corresponding to reduced monomials xα, so the rows of M?
following polynomial system:
⎧
⎩
For each polynomial of (3.14), we consider its term whose monomial is
ˆS2=?xα: xα∈ S2, αi? m−1 for some 3 ? i ? n?,
(3.13)
λis the submatrix of the coefficient matrix Mλobtained by deleting all
λcorrespond to the
⎨
?xα/xm−1
?xα/xm−1
1
?ˆ F1−λ?xα/xm−2
?ˆ Fn−1−λ?xα/xm−2
1
?I(x) = 0
n−1
for all xα∈ˆS1,
···
n−1
?I(x) = 0 for all xα∈ˆSn−1.
(3.14)
·xm−2
t
for some xα∈ˆSi.
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|ˆS??| =
i=1
Denote
i=?xα: |α| = (n−1)(m−2)+1, α1= ··· = αi−1= 0, αi? 1, αj? m−1?
for all i = 1,2,...,n−1, j = i +1,...,n. Then
G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
1225
which implies that its coefficient is an element of M?
there exists λ in each row of M?
Perform the same row operation on the matrix M?
we obtain a new matrix, denoted by ¯ M?.
LetˆS?=?n−1
ˆS?
i
and
ˆS??=?xα: |α| = (n−1)(m−2)+1, αi? 1, αj? m−1 for some 1 ? i < j ? n?.
λand contains a linear term of λ. Hence,
λ.
λas that on Mλin the proof of Theorem 3.1,
i=1ˆS?
i, where
i=?xα: xαxm−2
∈ˆSi
?,i = 1,2,...,n−1,
(3.15)
It is clear that
ˆSi
xm−2
i
=ˆS?
i
for all i = 1,...,n−1.
Moreover, similarly to the proof of Theorem 3.1, we have thatˆS?=ˆS??. Denote by r?the number
of entries of the setˆS??. Then, it is easy to observe that there are r?rows containing λ in ¯ M?and
r?= dM?. This completes the proof.
We now compute |ˆS??|.
2
Theorem 3.3.
|ˆS??| =
n−2
?
k=1
(−1)k−1
n−k
?
i=1
?n−i
k
?
·
?(n−1−k)(m−1)+1−i
n−i
?
.
Proof. Denote
Pi=?xα: |α| = (n−1)(m−2)+1, α1= ··· = αi−1= 0,
αi? 1, αj? m−1 for some i < j ? n?
for all i = 1,2,...,n−1. Then
n−1
?
Hence,
ˆS??=
i=1
Pi
and
Pi1∩Pi2= ∅
if i1?= i2.
n−1
?
|Pi|.
(3.16)
Pj
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G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
|Pi| =
n
?
n−i
?
j=i+1
??Pj
i
??−
?
i+1?j1<j2?n
??Pj1
i∩Pj2
i
??+
?
i+1?j1<j2<j3?n
??Pj1
i∩Pj2
i∩Pj3
i
??−···
(3.17)
=
k=1
(−1)k−1
?
i+1?j1<j2<···<jk?n
??Pj1
i∩Pj2
i∩···∩Pjk
i
??.
From the combinatory theory, it is clear that
?((n−1)(m−2)+1)−m+n−i
for j = i +1,...,n, and
??Pj1
=
??Pj
i
??=
n−i
?
=
?(n−2)(m−1)+1−i
n−i
?
i∩···∩Pjk
i
??=
?((n−1)(m−2)+1)−(k(m−1)+1)+n−i
?(n−1−k)(m−1)+1−i
n−i
?
n−i
?
,
(3.18)
where k = 1,...,n−i and i +1 ? j1< ··· < jk? n.
By (3.16)–(3.18), we have that
|ˆS??| =
n−1
?
n−1
?
n−1
?
n−1
?
i=1
n−i
?
n−i
?
n−i
?
k=1
(−1)k−1
?
?
?
?
i+1?j1<···<jk?n
??Pj1
?(n−1−k)(m−1)+1−i
?(n−1−k)(m−1)+1−i
?(n−1−k)(m−1)+1−i
i∩···∩Pjk
i
??
=
i=1
k=1
(−1)k−1
i+1?j1<···<jk?n
?n−i
?n−i
n−i
?
=
i=1
k=1
(−1)k−1
k
·
n−i
?
?
=
k=1
(−1)k−1
n−k
?
i=1
k
·
n−i
.
(3.19)
Note that, if k = n−1, then
?(n−1−k)(m−1)+1−i
By (3.19) and (3.20), we have the desired result.
n−i
?
= 0.
(3.20)
2
Theorem 3.4. Assume that A is a universal even order tensor. Then the degree of φ(λ) is given
by
d =
n−1
?
k=0
(m−1)k=
?n,
if m = 2,
otherwise.
(m−1)n−1
m−2
,
(3.21)
Furthermore, if A is a real even order tensor, then the above number d is an upper bound of the
degree of φ(λ).
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Hence, its E-characteristic polynomial is given by
φ(λ) = ±det(M4)
det(M?
Its E-eigenvalues are all Z-eigenvalues. They are λ = 1 (three multiple), 1/2 (six multiple) and
1/3 (four multiple). Totally, it has 13 Z-eigenvalues. Hence, when m = 4, the upper bound m2−
m+1 = 13 is attainable, i.e., d(4,3) = 13.
G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
1227
Proof. By (3.11) and Theorems 3.1–3.3, the degree of φ(λ) is given by
?(n−1)(m−1)+1
n−2
?
By induction, we have (3.21). But for a real even order tensor A, the leading coefficient cdof
the E-characteristic polynomial φ(λ) may be zero. In this case, deg(φ) < d, which follows the
second statement.
2
It is clear that the upper bound given by Theorem 3.4 is much smaller than that given by
Theorem 2.1 when m,n ? 3 and m is even.
d =
n−1
?
−
k=1
(−1)k−1
n−k
?
i=1
?n−i
k
?
·
?(n−1−k)(m−1)+1−i
n−i
?
.
Corollary 3.1.
d(2,n) = n.
Proof. This follows from (3.21) directly.
2
Corollary 3.2. Assume that m is even and m ? 2. Then
d(m,2) ? m.
Proof. This also follows from (3.21) directly.
2
The above two corollaries are the same as the corresponding contents of Theorem 2.1. In fact,
we have d(m,2) = m for all m ? 2. The following corollary is sharper than the corresponding
content of Theorem 2.1.
Corollary 3.3. Assume that m is even and m ? 2. Then
d(m,3) ? m2−m+1.
In particular, we have d(4,3) = 13.
Proof. The first statement also follows from (3.21) directly.
Let A be a 4th order 3-dimensional unit tensor, i.e., a1111= a2222= a3333= 1 and other
entries are zero. We have
det(Mλ) = (1−3λ)4(−1+λ)10(−1+2λ)7,
det?M?
λ
?= (−1+λ)7(−1+2λ).
4)= ±(1−3λ)4(−1+λ)3(−1+2λ)6.
2
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[15] L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal.
Appl. 21 (2000) 1253–1278.
[16] L. De Lathauwer, B. De Moor, From matrix to tensor: Multilinear algebra and signal processing, in: J. McWhirter
(Ed.), Mathematics in Signal Processing IV, Selected papers presented at 4th IMA Int. Conf. on Mathematics in
Signal Processing, Oxford University Press, Oxford, UK, 1998, pp. 1–15.
[17] L.-H. Lim, Singular values and eigenvalues of tensors: A variational approach, Proceedings of the 1st IEEE Interna-
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2005, pp. 129–132.
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G. Ni et al. / J. Math. Anal. Appl. 329 (2007) 1218–1229
We conjecture that the upper bound given in Theorem 3.4 is attainable and thus gives the exact
value of d(m,n). We also conjecture that Theorem 3.4 also holds when m is odd.
The following is an example that the degree of the E-characteristic polynomial of a 4th order
3-dimensional supersymmetric tensor is strictly less than 13.
Example 3.1. Let A be a 4th order 3-dimensional supersymmetric tensor with a2222= a3333= 1,
a1122= 1/6 and other entries are zero. We have
det(Mλ) =(1−2λ)2(−1+λ)8λ8
16384
det?M?
Hence, its E-characteristic polynomial is given by
φ(λ) = ±(1−2λ)2(−1+λ)6λ3
4096
It is clear that the degree of φ(λ) is less than 13.
,
λ
?=(−1+λ)2λ5
4
.
.
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