A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

Yiyong LI, Qingzhi YANG, Yuning YANG
Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract We give a new definition of geometric multiplicity of eigenvalues of
tensors, and based on this, we study the geometric and algebraic multiplicity
of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with
modulus ρ(A ) have the same geometric multiplicity. We also prove that twodimensional
nonnegative tensors’ geometric multiplicity of eigenvalues is equal
to algebraic multiplicity of eigenvalues.
Keywords Irreducible tensor, Perron-Frobenius theorem, geometrically simple
MSC 15A18, 15A69, 74B99
1 Introduction
Eigenvalue problems of nonnegative tensors, especially, the largest eigenvalue
of a nonnegative tensor, have attracted special attention in the recent years, see
[4,5,7,9,12,14,15]. In particular, the Perron-Frobenius theorem for nonnegative
tensors is related to measuring higher order connectivity in linked objects and
hypergraphs [1,2]. From the matrix theory, we know that the largest eigenvalue
of a nonnegative irreducible matrix is geometrically simple and the
eigenvalues with modulus ρ(A) are equally distributed on the spectral circle.
But the geometric and algebraic multiplicity for a higher order tensor becomes
complicated.
In the literature, the geometric simplicity of the largest eigenvalue (here
we call it the spectral radius) of a nonnegative irreducible tensor was studied
by Chang et al. [4], where some conditions were proposed to ensure the real
geometric simplicity of the spectral radius. Pearson [13] defined essentially
positive tensors and proved that the spectral radius of an essentially positive
Received April 4, 2014; accepted July 4, 2014
Corresponding author: Qingzhi YANG, E-mail: qz-yang@nankai.edu.cn
1124 Yiyong LI, et al.
tensor with order even is real geometrically simple and the modulus of any other
eigenvalue of the essentially positive tensor with even order is smaller than the
unique positive eigenvalue. Chang et al. [6] showed the modulus of any other
eigenvalue of the primitive tensor is smaller than the unique positive eigenvalue.
In [19], it was proved that the spectral radius is complex geometrically simple
for any positive tensors. The real geometric simplicity of an even order nonnegative
irreducible tensor can be deduced from a conclusion of [20]. In [3], the
real geometric simplicity of the spectral radius of an even order nonnegative
irreducible tensor was proved in totally different way from that of [20].
This paper is organized as follows. In Section 2, we recall some definitions
and well-known results. In Section 3, we give the new definition of geometric
multiplicity of tensor’s eigenvalues, and based on this, we study the geometric
and algebraic multiplicity of irreducible tensors’ eigenvalues. We get the result
that the eigenvalues with modulus ρ(A ) have the same geometric multiplicity.
The eigenvalues λe2πij/k (j = 1, . . . , k) of an order m dimension n nonnegative
weakly irreducible tensor have the same algebraic multiplicity. Although this
result can be deduced from [16], the idea of our analysis is totally different
from that of [16]. In Section 4, we prove that for two-dimensional nonnegative
tensors, the geometric multiplicity is equal to the algebraic multiplicity.
We first add a comment on the notation that is used in the sequel.
Vectors are written as lowercase letters (x, y, . . .), matrices correspond to italic
capitals (A,B, . . .), and tensors are written as calligraphic capitals (A ,B, . . .).
The entry with row index i and column index j in a matrix A, (A)ij , is
symbolized by aij (also, (A )i1···ip,j1···jq = ai1···ip,j1···jq ). The symbol | · | used on
a matrix A (resp. tensor A ) means that (|A|)ij = |aij | (resp. (|A |)i1···ip,j1···jq =
|ai1···ip,j1···jq
|). Rn
+ (resp. Rn
++) denotes the cone {x ∈ Rn | xi (resp. >) 0, i =
1, . . . , n}. The symbol A (>, ,<)B means that aij (>, ,<) bij for every
i, j and it is the same for rectangular tensors.
2 Preliminaries
First, we recall some known definitions and results, and then we give some new
definitions and remarks.
A tensor is a multidimensional array, and a real order m dimension n tensor
A consists of nm real entries:
ai1···im
∈ R,
where ij = 1, . . . , n for j = 1, . . .,m. Like the square matrix, setting this class
of tensors can be regarded as “square” tensors. If there are a complex number λ
and a nonzero complex vector x that are solutions of the following homogeneous
polynomial equations:
A xm−1 = λx[m−1],
then λ is called the eigenvalue of A and x the eigenvector of A associated with
New definition of geometric multiplicity of eigenvalues of tensors 1125
λ, where A xm−1 and x[m−1] are vectors, whose ith component are
(A xm−1)i =
n
i2,...,im=1
ai,i2···imxi2
· · · xim, (x[m−1])i = xm−1
i ,
respectively. This definition was introduced by Qi [14] where he assumed that
A is an order m dimension n symmetric tensor and m is even. If λ and x are
restricted in the real field, then (λ, x) is called an H-eigenpair. Independently,
Lim [11] gave such a definition but restricted x to be a real vector and λ to be a
real number. Here, we follow the definition due to Chang et al. [4], where they
gave the general definition as above.
In the following analysis, the notions below will be used.
Definition 1 [19, Definition 2.2] The spectral radius of tensor A is defined
as
ρ(A) = max{|λ| : λ is an eigenvalue of A }.
Definition 2 [11] A tensor C = (ci1
· · · cim) of order m dimension n is called
reducible, if there exists a nonempty proper index subset I ⊂ {1, . . . , n} such
that
ci1···im = 0, ∀ i1 ∈ I, ∀ i2, . . . , im ∈ I.
If C is not reducible, then we call C irreducible.
Definition 3 [6, Definition 2.6] An order m dimension n nonnegative
irreducible tensor A is called primitive, if TA does not have a nontrivial
invariant set S on ∂P. ({0} is the trivial invariant set.)
Definition 4 [8, Definition 2.1] A tensor C = (ci1
· · · cim) of order m
dimension n is called essentially positive, if C xm−1 > 0 for any nonzero x 0.
Definition 5 [10, Definition 2.1] A nonnegative matrix M(A ) is called the
majorization associated to nonnegative tensor A , if the (i, j)-th element of
M(A ) is defined to be aij...j for any i, j ∈ 1, . . . , n. A is called weakly positive
if [M(A )]ij > 0 for all i = j.
Definition 6 A tensor D(A ) = (bi1···im) is called induced to the tensor A =
(ai1···im), if
bi1···im =

ai1···im, i2 = · · · = im = i, i = 1, . . . , n,
0, otherwise,
for any i1 ∈ {1, . . . , n}. A is called a degenerated tensor, if A =D(A ).
In the next section, we will give some results about degenerated tensor.
Definition 7 [10, Definition 2.2] Suppose that A is a nonnegative tensor of
order m and dimension n.
1126 Yiyong LI, et al.
(1) We call a nonnegative matrix G(A ) the representation associated to
the nonnegative tensor A , if the (i, j)-th element of G(A ) is defined to be the
summation of A{ii2...im} with indices {i2 · · · im} j.
(2) We call the tensor A weakly reducible, if its representation G(A ) is a
reducible matrix, and weakly primitive, if G(A ) is a primitive matrix. If A is
not weakly reducible, then it is called weakly irreducible.
Definition 8 An order m dimension n tensor A is called strongly irreducible
(resp. primitive) tensor, if D(A ) is irreducible (resp. primitive) tensor.
Lemma 1 D(A ) is irreducible (resp. primitive) tensor if and only if M(A )
is irreducible (resp. primitive) matrix.
Proof It is obvious by Definitions 5 and 6.
Definition 9 [13, Definition 3.4] For any vector x ∈ Rn , we define
(x[1/(m−1)])i = x1/(m−1)
i .
Let A and B be two order m dimension n nonnegative tensors. Let
ω := (A xm−1)[1/(m−1)] ∈ Rn.
We define the composite of the tensors for x ∈ Rn to be the function (not
necessarily a tensor) (B ◦ A )x = Bωm−1.
Definition 10 [4, Definition 3.1] Let λ be an eigenvalue of
A xm−1 = λx[m−1].
We say that λ has geometric multiplicity q, if the maximum number of linearly
independent eigenvectors corresponding to λ equals q. If q = 1, then λ is called
geometrically simple.
We follow [14] to define the characteristic polynomial ψ(λ) of λ by
ψ(λ) = Res((A xm−1)1 − λx[m−1]
1 , . . . , (A xm−1)1 − λx[m−1]
n ),
where Res(P1, . . . , Pn) is the resultant of n homogeneous polynomials P1, . . . , Pn.
For each λ, such ψ(λ) is unique up to an extraneous factor.
Definition 11 [4, Definition 3.1] Let λ be an eigenvalue of λ. We say that λ
has algebraic multiplicity p, if λ is a root of ψ(λ) of multiplicity p. And we call
λ an algebraically simple eigenvalue, if p = 1.
Theorem 1 [18, Theorem 3.6] Let A and B be order m dimension n tensors
satisfying |B| A , and let A be weakly irreducible. Let β be an eigenvalue of
B. Then
(1) |β| ρ(A );
New definition of geometric multiplicity of eigenvalues of tensors 1127
(2) if β = ρ(A )eiϕ, then there exists a diagonal matrix D with modular-1
diagonal entries such that
A = e−iϕB · D
−(m−1) ·
m−1
D· · · D.
Theorem 2 [18, Theorem 3.7] Let A be an order m dimension n nonnegative
weakly irreducible tensor. Suppose (λ, y) is an eigen-pair of A with modulus of
λ equal to ρ(A ), i.e., |λ| = ρ(A ). Then |y| is the unique positive eigenvector
corresponding to ρ(A ).
Theorem 3 [19, Theorem 3.1] Let A be a nonnegative irreducible tensor and
it has k different eigenvalues with modulus ρ(A ). Then these eigenvalues are
ρ(A )e2πij/k, j = 1, . . . , k.
Theorem 4 [6, Theorem 4.5] Let A be primitive. If λ is an eigenvalue with
|λ| = ρ(A ), then λ = ρ(A ).
Theorem 5 [13, Theorem 3.7] If A is an order m dimension n nonnegative,
essentially positive tensor with even m, then the modulus of any other
eigenvalue of A is smaller than the unique positive eigenvalue; moreover, the
unique positive eigenvalue is real geometrically simple.
3 Geometric multiplicity of tensors’ eigenvalue
As we have known, for a matrix, the set of eigenvectors corresponding to an
eigenvalue λ is linear space, so we can call the maximum number of linearly
independent eigenvectors the geometric multiplicity of λ. If a set is not linear
space, the dimension is not meaning for this set. But if the set have property
similar to linear space, we should give definitions. By Theorem 8 below, the set
of eigenvectors corresponding to ρ(A ) for an irreducible tensor have property
of
G = G1 ∪ · · · ∪ Gp, p (m − 1)r(n−1),
where
Gj = {kx = 0 | x[(m−1)r] = |x|[(m−1)r], k ∈ C }, Gi ∩ Gj = ∅, 1 i, j p.
Thus, we give a new definition of geometric multiplicity.
Before we proceed with the new definition of geometric multiplicity, we use
three examples to preview some of the main ideas of the definition.
(i)
G = {x = 0 | x = k(k1x1 + k2x2), k ∈ C, k1, k2, k1 + k2 = 0 or 1}.
We think that the multiplicity of G is 2 and B = {x1, x2} ⊂ G is the set of
extreme directions.
1128 Yiyong LI, et al.
(ii)
G = {x = 0 | x = k[k1x1 + k2x2 + k3(x1 + 3x2)],
k ∈ C, k1, k2, k3, k1 + k2 + k3 = 0 or 1}.
We think that the multiplicity of G is 3 and B = {x1, x2, x1 + 3x2} ⊂ G is the
set of extreme directions.
(iii)
G = {x = 0 | x = k1x1 + k2(k21x2 + k22x3),
k1, k2 ∈ C, k21, k22, k21 + k22 = 0 or 1}.
We think that the multiplicity of G is 3 and B = {x11, x21, x22} ⊂ G is the set
of extreme directions.
In order to make the definition of geometric multiplicity easy to be
understood, we give the following definition and corollary.
Definition 12 If
A ∩ B = ∅, C= {k1x + k2y = 0 | ∀x ∈ A, ∀ y ∈ B, k1, k2 ∈ C }.
Then we say that C is the general direct sum of A and B, and write it as
C = A ⊕ B.
Corollary 1 We have

x = 0

x =
i1
i=1
ki

j1
j=1
kijxij

,
j1
j=1
kij = 1, kij = 0 or 1, xij ∈ B, ki ∈ C

=

i1
i=1
ji
j=1
Aij

,
where
Aij = {y = 0 | y = kixij = 0, xij ∈ B, ki ∈ C }.
Proof For any
x ∈

ki

j1
j=1
kijxij

= 0

j1
j=1
kij = 1, ki ∈ C

, (3.1)
we have x = kixij . Then x ∈ ∪ji
j=1Aij . And for x ∈ ∪ji
j=1Aij , we have x = kixij ,
and then we get (3.1). Hence,

ki

j1
j=1
kijxij

= 0

j1
j=1
kij = 1, ki ∈ C

=
ji
j=1
Aij .
And by Definition 12, we have

x = 0

x =
i1
i=1
kixi, xi ∈
ji
j=1
Aij

, ki ∈ C

=

i1
i=1
ji
j=1
Aij

,
New definition of geometric multiplicity of eigenvalues of tensors 1129
which gives the desired result.
Definition 13 Let
G = {x = 0 | A xm−1 = λx[m−1]}
be the set of eigenvectors corresponding to eigenvalue λ of the tensor A . If
G =

i1
i=1
ji
j=1
Aij

, Aij = {y | y = kijxij = 0, xij ∈ B, kij ∈ C }
(or
G =

x = 0

x =
i1
i=1
ki

j1
j=1
kijxij

,
j1
j=1
kij = 1,
kij = 0 or 1, xij ∈ B, ki ∈ C

by the Corollary 1). If q is the minimum of
i1
i=1 ji, then we say that λ has
geometric multiplicity q and we call B = {x1, . . . , xq} ⊂ G the generalized base
corresponding to λ; if q = 1, then λ is called geometrically simple.
Remark Definition 13 is given based on the geometric multiplicity of nonnegative
irreducible tensors and two-dimensional nonnegative tensors. Although
we cannot find a counter-example for the unreasonable of the definition for nonnegative
tensors, we should use the definition for the nonnegative irreducible
tensors and two-dimensional nonnegative tensors only. From Theorems 7 and 8
below, we can get that geometric multiplicity of λ with modulus ρ(A ) in
Definition 13 (m − 1)r(n−1). From Theorem 16 below, for an order m
dimension two nonnegative weakly irreducible tensor, the geometric multiplicity
of λ with modulus ρ(A ) is equal to the algebraic multiplicity of it. The
difference from the definition before is that the extreme direction corresponding
to λ can be linearly dependent. For example,
G = G1 ∪ · · · ∪ Gq, q (m − 1)r(n−1),
where
Gj = {kx = 0 | x[(m−1)r] = |x|[(m−1)r ], k ∈ C}, Gi ∩ Gj = ∅, 1 i, j p,
and thus,
G =

x = 0

x = k
q
i=1
kixi,
q
i=1
ki = 1, ki = 0 or 1, xi ∈ B, k ∈ C

.
Then the extreme direction of G is B = {x1, . . . , xq}. If x1, . . . , xq are linearly
dependent and
(x1, . . . , xq) = (x1, . . . , xr)(αT1
, αT2
· · · αTq
),
1130 Yiyong LI, et al.
then x1, . . . , xr cannot be worked out G without αTr
+1, αTr
+1
· · · αTk
. And the
geometric multiplicity in Definition 11 is r, but the geometric multiplicity in
Definition 13 is q. Obviously, r q. Then we will give a general result about
these two definitions.
Corollary 2 Geometric multiplicity in Definition 11 complex geometric
multiplicity in Definition 13.
Proof Suppose that
B = {xij | 1 i i1, 1 j ji} ⊆ G
is the generalized base corresponding to λ and B0 = {x1, . . . , xq} is the
maximum linearly independent group of B. Then q
i1
i=1 ji and
B0 ⊂ B
⊂ G
=

x = 0

x =
i1
i=1
ki

j1
j=1
kijxij

,
j1
j=1
kij = 1,
kij = 0 or 1, xij ∈ B, ki ∈ C

⊂

x = 0

x =
q
j=1
kixi, ki ∈ C

.
So q is geometric multiplicity in Definition 11. And
i1
i=1 ji is geometric
multiplicity in Definition 13. Hence, geometric multiplicity in Definition 11
complex geometric multiplicity in Definition 13.
Remark If
B0 = {x1, . . . , xq} ⊂ G1 ⊂ G2 ⊂

x = 0

x =
q
j=1
kixi, ki ∈ C

,
then G1 and G2 have the same geometric multiplicity in Definition 11. But this
is different in Definition 13. For example,
B0 = {x1, x2}
⊂ G1
= {x = 0 | x = k[k1x1 + k2x2], k ∈ C, k1, k2, k1 + k2 = 0 or 1}
⊂ G2
= {x = 0 | x = k[k1x1 + k2x2 + k3(x1 + 3x2)], k ∈ C ,
k1, k2, k3, k1 + k2 + k3 = 0 or 1}
⊂ {x = 0 | x = k1x1 + k2x2, k1, k2 ∈ C },
New definition of geometric multiplicity of eigenvalues of tensors 1131
G1 and G2 both have the geometric multiplicity 2 in Definition 11, but G1
has the geometric multiplicity 2 in Definition 13 and G1 has the geometric
multiplicity 3 in Definition 13.
Theorem 6 Let A be an order m dimension n nonnegative strongly
irreducible tensor, and let G be the set of eigenvectors corresponding to ρ(A ).
If x ∈ G, then
x[m−1] = eiθ|x|[m−1].
Moreover, ρ(A ) is real geometrically simple when m is even.
Proof Because A is a nonnegative strongly irreducible tensor, M(A ) is a
nonnegative irreducible matrix. Suppose that
∃ x ∈ A, x[m−1] = eiθ|x|[m−1], y=

|x1|,
|x1|
x1
x2, . . . ,
|x1|
x1
xn

.
Then y[m−1] = |y|[m−1], and
A ym−1 = ρ(A )y[m−1], A |y|m−1 = ρ(A )|y|[m−1].
Suppose
(|y|[m−1] − y[m−1])j

= 0, 1 j i
= 0, i+ 1 j n.
Then
Re(|yi2
| · · · |yim
| − yi2
· · · yim) > 0, i+ 1 i2, . . . , im n,
Re(|yj |[m−1] − y[m−1]
j ) > 0, i+ 1 j n.
So
Re[ρ(A )(|y|[m−1] − y[m−1])] = Re(A |y|m−1 − A ym−1)
Re[M(A )(|y|[m−1] − y[m−1])]
= M(A )Re(|y|[m−1] − y[m−1]).
And we also have
(A |y|m−1 − A ym−1)k = 0, 1 k i.
Thus, we can get
ak,j···j = 0, 1 k i, i+ 1 j n.
Hence, D(A ) is a nonnegative reducible tensor, which contradicts that D(A )
is a nonnegative reducible tensors. Then
x[m−1] = eiθ|x|[m−1].
Clearly, ρ(A ) is real geometrically simple when m is even.
1132 Yiyong LI, et al.
Theorem 7 Let A be an order m dimension n nonnegative irreducible tensor,
and let G be the set of eigenvectors corresponding to ρ(A ). If x ∈ G, then
x[(m−1)r] = eiθ|x|[(m−1)r],
where θ relies on x, and r is the number of PTM(A)P irreducible blocks.
Moreover, ρ(A ) is real geometrically simple with m even.
Proof Suppose
PTM(A)P = (Dij)1 i,j r,
where P is an n × n permutation matrix, Dii, i = 1, . . . , r, are irreducible
matrices. Let
z[m−1] = PTx[m−1] = (z[m−1]
1 , . . . , z[m−1]
r )T.
Since
ρ(A )(|x|[m−1] − xm−1) = A |x|m−1 − A xm−1,
we have
Re[ρ(A )(|x|[m−1] − x[m−1])] Re[M(A )(|x|[m−1] − x[m−1])].
Thus,
Re[ρ(A )(PT|x|[m−1] − PTx[m−1])] Re[PTM(A )P(PT|x|[m−1] − PTx[m−1])],
Re[ρ(A )(|z|[m−1] − z[m−1])] Re[PTM(A )P(|z|[m−1] − z[m−1])].
Hence,
Re[ρ(A )(|z|[m−1]
i
− z[m−1]
i )] Re[Dii(|z|[m−1]
i
− z[m−1]
i )], 1 i r.
Then, by the proof of Theorem 6, we can get
z[m−1]
i = eiθi |z|[m−1]
i , 1 i r.
Suppose that
z = PTx = (z1, . . . , zr), z[m−1]
i = eiθi |z|[m−1]
i , 1 i r,
and z1 contains p elements. Next, we will prove that if x ∈ G, then
x[(m−1)r] = eiθ|x|[(m−1)r]
by inductive method.
Case 1 When r = 2, let
y = z = (z1, z2), ξ= PT(1, . . . , n)T.
New definition of geometric multiplicity of eigenvalues of tensors 1133
Then yi = xξi . Since A is a nonnegative irreducible tensor, ∃ ai1···im > 0, i2, . . . ,
i
m ∈ {ξ1, . . . , ξp}, p is the side of z1, and i1 ∈ {ξp+1, . . . , ξn}. Then we have
n
j2,...,jm=1
a i1j2···jm
xj2
· · · xjm = eiθ2ρ(A )|x i1
|m−1,
n
j2,...,jm=1
a i1j2···jm
|xj2
| · · · |xjm
| = ρ(A )|x i1
|m−1,
n
j2,...,jm=1
a i1j2···jm
|xj2
| · · · |xjm
| =

n
j2,...,jm=1
a i1j2···jm
xj2
· · · xjm

,
a i1··· im
x i2
· · · x im
= eiθ2a i1··· im
|x i2
| · · · |x im
|,
ei(m−1)θ1am−1
i
1··· im
|x i2
|m−1 · · · |x im
|m−1 = ei(m−1)θ2am−1
i
1··· im
|x i2
|m−1 · · · |x im
|m−1.
Since a i1··· im
> 0, we have
ei(m−1)θ1 = ei(m−1)θ2 .
Thus,
z[(m−1)2] = eiθ|z|[(m−1)2].
Then
x[(m−1)2] = eiθ|x|[(m−1)2].
Case 2 When
r = k, z = PTx = (z1, . . . , zk), z[m−1]
i = eiθi |z|[m−1]
i , 1 i k,
we have
x(m−1)k = eiθ|x|(m−1)k
.
Case 3 When r = k + 1, let
y = z = (z1, . . . , zk+1), z[m−1]
i = eiθi |z|[m−1]
i , 1 i k + 1,
ξ = PT(1, . . . , n)T.
Then yi = xξi . Since A is a nonnegative irreducible tensor, ∃ ai1···im > 0, i2, . . . ,
i
m ∈ {ξ1, . . . , ξp}, p is the side of z1, i1 ∈ {ξp+1, . . . , ξn}, and x i1
is the
component of zj . Then we have
n
j2,...,jm=1
a i1j2···jm
xj2
· · · xjm = eiθjρ(A )|x i1
|m−1,
n
j2,...,jm=1
a i1j2···jm
|xj2
| · · · |xjm
| = ρ(A )|x i1
|m−1,
1134 Yiyong LI, et al.
n
j2,...,jm=1
a i1j2···jm
|xj2
| · · · |xjm
| =

n
j2,...,jm=1
a i1j2···jm
xj2
· · · xjm

,
a i1··· im
x i2
· · · x im
= eiθja i1··· im
|x i2
| · · · |x im
|,
ei(m−1)θ1am−1
i
1··· im
|x i2
|m−1 · · · |x im
|m−1 = ei(m−1)θj am−1
i
1··· im
|x i2
|m−1 · · · |x im
|m−1.
Since ai1···im > 0, we have
ei(m−1)θ1 = ei(m−1)θj .
Thus,
z[(m−1)2]
1 = eiθ|z1|[(m−1)2], z[(m−1)2]
j = eiθ|zj |[(m−1)2].
Let
w = (w1, . . . ,wk), wi =
⎧⎪⎨
⎪⎩
(z1, zj ), i= 1,
zi, 2 i j − 1,
zi+1, j+ 1 i k.
Then
r = k, w = PTx = (w1, . . . ,wk), w[m−1]2
i = eiθi |z|[m−1]2
i , 1 i k.
Thus, we have
x(m−1)k+1 = eiθ|x|(m−1)k+1
.
Moreover, ρ(A ) is real geometrically simple when m is even.
Corollary 3 If
G0 = {x | x[(m−1)r] = |x|[(m−1)r]}, Gθ = {x | x[(m−1)r] = eiθ|x|[(m−1)r]},
then
Gθ = {eiθ/(m−1)r
x | x ∈ G0}.
Moreover, if A is an order m dimension n nonnegative irreducible tensor and
G is the set of eigenvectors corresponding to ρ(A ), then the side of G is not
more than (m − 1)r up to a multiplicative constant.
Proof For any x ∈ Gθ, we have
x[(m−1)r] = eiθ|x|[(m−1)r].
Then
x = (ei θ+2j1π
(m−1)r |x1|, . . . , ei θ+2jnπ
(m−1)r |xn|)
= ei θ
(m−1)r (ei 2j1π
(m−1)r |x1|, . . . , ei 2jnπ
(m−1)r |xn|).
Since
(ei 2j1π
(m−1)r |x1|, . . . , ei 2jnπ
(m−1)r |xn|) ∈ G0,
New definition of geometric multiplicity of eigenvalues of tensors 1135
we have
Gθ = {ei θ
(m−1)r x | x ∈ G0}.
The side of G0 is not more than (m − 1)r, so the side of G is not more than
(m − 1)r up to a multiplicative constant by Theorem 7.
Lemma 2 Let A be an order m dimension n nonnegative irreducible tensor,
and let G be the set of eigenvectors corresponding to ρ(A ). If x, y, k1x + k2y ∈
G, k1k2 = 0, then
xj
yj
= x1
y1
or
xj
yj
= x1
y1
eiθ,
and
eiθ k1
k2
= k1
k2
.
Proof Since x, y ∈ G, |y| = k|x| by Theorem 2. Since k1x + k2y ∈ G, we have
|k1x + k2y| = p|x|.
Thus,
p|xj | = |k1xj + k2yj |.
Hence,
p
|k1| =

1 + k2
k1
yj
xj

,
1

j

n,
and the desired result follows.
Theorem 8 Let A be an order m dimension n nonnegative irreducible tensor,
and let G be the set of eigenvectors corresponding to ρ(A ). Then
G = G1 ∪ · · · ∪ Gp, p (m − 1)r(n−1), (3.2)
where
Gj = {kx = 0 | x[(m−1)r] = |x|[(m−1)r], 0 = k ∈ C}, Gi∩Gj = ∅, 1 i, j p.
Hence, {x1, . . . , xp} ⊂ G is the generalized base of ρ(A ) and ρ(A ) has
geometric multiplicity p.
Proof We get (3.2) by Theorem 7 and Corollary 3. From (3.2), we can get the
result that ∀ x ∈ G, x = p0
k
i=1 pixi, xi ∈ Gi, 1 i p, p0 ∈ C, {pi, 1 i
p}, does not have more than one nonzero element. Suppose that the geometric
multiplicity of ρ(A ) is k0 < p. Then we have
G = B1 ∪ · · · ∪ Bk0,
where
Bj = {kx = 0 | x[(m−1)r] = |x|[(m−1)r], 0 = k ∈ C}, Bi∩Bj = ∅, 1 i, j k0,
1136 Yiyong LI, et al.
by Lemma 2. Then we get that there exist i0 and j0 such that 1 i0, j0
p, Gi0 = Gj0 , which contradicts that Gi ∩ Gj = ∅, 1 i, j p. So we get the
conclusion that for 0 = xj ∈ Gj , 1 j p, {x1, . . . , xk} ⊂ G is the generalized
base of ρ(A ) and ρ(A ) has geometric multiplicity p by Definition 13.
Lemma 3 Suppose that G1 is the set of eigenvectors corresponding to λ1 and
G2 is the set of eigenvectors corresponding to λ2. If G1 is isomorphic to G2
and the geometric multiplicity of λ1 is finite, then λ1 and λ2 have the same
geometric multiplicity.
Proof Because G1 is isomorphic to G2 and the geometric multiplicity of λ1
is finite, we can suppose that {x1, . . . , xk} ⊂ G1 is the base of G1 and there
exists isomorphism ϕ, which satisfy {ϕ(x1), . . . , ϕ(xp)} ⊂ G2 and G2 can not
only be linear represented by {ϕ(x1), . . . , ϕ(xp)}, but also can be worked out
by {ϕ(x1), . . . , ϕ(xp)}. Thus, the geometric multiplicity of λ2 is finite and the
geometric multiplicity of λ2 is not more than the geometric multiplicity of λ1.
Thus, the geometric multiplicity of λ1 is not more than the geometric
multiplicity of λ2 by the same proof. Therefore, λ1 and λ2 have the same
geometric multiplicity.
Theorem 9 Let A be an order m dimension n nonnegative irreducible tensor.
Suppose that ρ(A ) and ρ(A )eiθ are eigenvalues of A . Then these eigenvalues
have the same geometric multiplicity.
Proof Suppose
A ym−1 = ρ(A )eiθy[m−1], A xm−1 = ρ(A )x[m−1],
D = diag
y1
|y1|, . . . ,
yn
|yn|

.
Then
A = e−iθA · D
−(m−1) ·
m−1
D· · · D.
Thus,
n
i2,...,im=1
aii2···imxi2
· · · xim
= e−iθ
n
i2,...,im=1
ai,i2···im
yi
|yi|
−(m−1) yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim
= e−iθ
yi
|yi|
−(m−1)
n
i2,...,im=1
ai,i2···im
yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim
= ρ(A )xm−1
i ,
and then
n
i2,...,im=1
ai,i2···im
yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim = ρ(A )eiθ
yi
|yi| xi
m−1
.
New definition of geometric multiplicity of eigenvalues of tensors 1137
Letting
z =
y1
|y1| x1, . . . ,
yn
|yn| xn

,
we have
A zm−1 = ρ(A )eiθz[m−1].
For
x = (x1, . . . , xn) ∈ G0, y= (y1, . . . , yn) ∈ G1,
where G0 is the set of eigenvectors corresponding to ρ(A ) and G1 is the set of
eigenvectors corresponding to ρ(A )eiθ, define project ϕ as follows:
ϕ(x) = z =
y1
|y1| x1, . . . ,
yn
|yn| xn

.
Then, ∀ x, x ∈ Ak, x = x , we have ϕ(x) = ϕ(x ). So ϕ is an injection from G0
to G1. In addition, for y ∈ G1, since
A ym−1 = ρ(A )eiθy[m−1],
we have
A ym−1 = ρ(A )e−iθy[m−1],
where y denotes the dual of y. Let
D1 = diag
y1
|y1
|, . . . ,
yn
|yn
|

.
Then
A = eiθA · D
−(m−1)
1
·
m−1
D1 · · ·D1.
Thus, for any z ∈ G1,
n
j2,...,jm=1
aij2···jmzj2
· · · zjm
= eiθ
n
j2,...,im=1
aij2···jm
yi
|yi
|
−(m−1) yj2
|yj2
|
· · ·
yjm
|yjm
|
· zj2
· · · zjm
= eiθ
yi
|yi
|
−(m−1)
n
j2,...,jm=1
ai,j2···im
yj2
|yj2
|
· · ·
yjm
|yjm
|
· zj2
· · · zjm
= ρ(A )eiθzm−1
i ,
and then
n
j2,...,jm=1
ai,j2···im
yj2
|yj2
|
· · ·
yjm
|yjm
|
· zj2
· · · zjm = ρ(A )
yi
|yi
| zi
m−1
.
1138 Yiyong LI, et al.
Letting
x =
y1
|y1
|

z1, . . . ,
yn
|yn
|

zn

,
we have
A xm−1 = ρ(A )x[m−1].
So for any z ∈ G1, there exists
x =
y1
|y1
|

z1, . . . ,
yn
|yn
|

zn

∈ G0
such that ϕ(x) = z, and then ϕ is a surjection. And we also have
ϕ(k1x+k2x
) =
y1
|y1| (k1x+k2x
)1, . . . ,
yn
|yn| (k1x+k2x
)n

= k1ϕ(x)+k2ϕ(x
).
So G0 is isometry to G1. Then ρ(A ) and ρ(A )eiθ have the same geometric
multiplicity by Theorem 8 and Lemma 3.
Theorem 10 Let A be an order m dimension n nonnegative irreducible
tensor, which has k different eigenvalues with modulus ρ(A ). Suppose that Gj
is the set of eigenvectors corresponding to ρ(A )e2πij/k, j = 1, . . . , k. Then these
eigenvalues have the same geometric multiplicity.
Proof Take θ = 2πj/k, j = 1, . . . , k. Then the conclusion follows immediately
from Theorem 9.
Theorem 11 Let A be an order m dimension n nonnegative irreducible
tensor, which has k different eigenvalues with modulus ρ(A ). Suppose that Gj
is the set of eigenvectors corresponding to λe2πij/k, j = 1, . . . , k, and y1 is an
eigenvector corresponding to ρ(A )e2πi/k. Then Gj, j = 1, . . . , k, are isometry
each other and
Gp =

z ∈ Cn

x ∈ Gk, z =
y11
|y11|
p
x1, . . . ,
y1n
|y1n|
p
xn

, p= 1, . . . , k−1.
Proof The conclusion is easily obtained from Theorem 9.
Theorem 12 Let A be an order m dimension n nonnegative irreducible
tensor, which has k different eigenvalues with modulus ρ(A ), and let Ap be the
set of eigenvectors corresponding to ρ(A )e2πip/k, p = 1, . . . , k. If x ∈ Gp, y ∈
Ak, then
zj =
y1
|y1|
j
x1, . . . ,
yn
|yn|
j
xn

∈ Ap, j∈ Z, p = 1, . . . , k.
Proof Suppose that
A xm−1 = ρ(A )e2πip/kx[m−1], A ym−1 = ρ(A )y[m−1],
D = diag
y1
|y1| , . . . ,
yn
|yn|

, D1 = diag
yj01
|yj01
|, . . . ,
yj0n
|yj0n
|

.
New definition of geometric multiplicity of eigenvalues of tensors 1139
Then
A = A · D
−(m−1) ·
m−1
D· · ·D = A · D
−(m−1)
1
·
m−1
D1 · · ·D1.
Thus,
n
i2,...,im=1
ai,i2···imxi2
· · · xim
=
n
i2,...,im=1
ai,i2···im
yi
|yi|
−(m−1) yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim
=
yi
|yi|
−(m−1)
n
i2,...,im=1
ai,i2···im
yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim
= ρ(A )e2πip/kxm−1
i ,
and then
n
i2,...,im=1
ai,i2···im
yi2
|yi2
|
· · · yim
|yim
|
· xi2
· · · xim = ρ(A )e2πip/k
yi
|yi| xi
m−1
.
Letting
z1 =
y1
|y1| x1, . . . ,
yn
|yn| xn

,
we have
A zm−1
1 = ρ(A )e2πip/kz[m−1]
1 .
Repeating the procedure above, we have
zj =
y1
|y1|
j
x1, . . . ,
yn
|yn|
j
xn

,
which satisfies
A zm−1
j = ρ(A )e2πip/kz[m−1]
j , j∈ N.
With the same progress, we can get
A zm−1
j = ρ(A )e2πip/kz[m−1]
j , j∈ N.
Thus, zj ∈ Gp, j ∈ Z, p = 1, . . . , k.
Corollary 4 Let A be an order m dimension n nonnegative irreducible
degenerated tensor, and let G be the set of eigenvectors corresponding to ρ(A ).
Then
complex geometric multiplicity of ρ(A ) = algebraic multiplicity of ρ(A )
= (m − 1)n−1.
1140 Yiyong LI, et al.
Proof Suppose that G is the set of eigenvectors of ρ(A ). Because A is a
nonnegative irreducible degenerated tensor, M(A ) is an irreducible matrix.
Then
ρ(A )x[m−1] = M(A )x[m−1] = ρ(A )x[m−1].
Thus,
x[m−1] = |x|[m−1], G= G1 ∪ G2 ∪ · · · ∪ G(m−1)n−1 ,
where
Gj = {kx = 0 | x[(m−1)r] = |x|[(m−1)r ], k ∈ C},
Gi ∩ Gj = ∅, 1 i, j (m − 1)n−1.
So we establish the complex geometric multiplicity corresponding to ρ(A) =
(m−1)n−1 by Theorem 8. And when λ is not the root of Res(M(A )−λI) = 0,
Res(A xm−1 − λx[m−1])
= Res(M(A )x[m−1] − λx[m−1])
= det(M(A ) − λI)(m−1)n−1Res(xm−1
1 , xm−1
2 , . . . , xm−1
n )
= det(M(A ) − λI)(m−1)n−1
.
Because the side of {λ | det(M(A ) − λI) = 0} is finite, for any λ, we have
Res(A xm−1 − λx[m−1]) = det(M(A ) − λI)(m−1)n−1
.
Then we can get the algebraic multiplicity of ρ(A ) = (m − 1)n−1, too. Thus,
we get the desired result.
In [4], Chang et al. pointed out that “By definition, we see complex
geometric multiplicity algebraic multiplicity, but not equal in general”.
However, by Definition 13, we can get
geometric multiplicity of ρ(A ) = algebraic multiplicity of ρ(A )
for some kinds of nonnegative tensors.
Corollary 5 Let A be an order m dimension n nonnegative irreducible tensor,
which has k different eigenvalues with modulus ρ(A ). Suppose that Gj is the set
of eigenvectors corresponding to λe2πij/k, j = 1, . . . , k. Then Gj, j = 1, . . . , k,
are isometry each other.
Proof Suppose
A ym−1 = ρ(A )eiθy[m−1], D= diag
y1
|y1| , . . . ,
yn
|yn|

.
Then
A = e−iθA · D
−(m−1) ·
m−1
D· · · D.
New definition of geometric multiplicity of eigenvalues of tensors 1141
Thus,
Res(A xm−1 − λx[m−1])
= Res(e−iθA · D
−(m−1) ·
m−1
D· · · Dxm−1 − λx[m−1])
= (e−iθ det(D))−(m−1)nRes(A (Dx)m−1 − eiθλ(Dx)[m−1])
= (e−iθ det(D))−(m−1)nRes(A zm−1 − eiθλz[m−1]),
where z = Dx. Suppose
ϕ(λ) = Res(A xm−1 − λx[m−1]),
ψ(eiθλ) = (e−iθ det(D))−(m−1)nRes(A zm−1 − eiθλz[m−1]).
Then
ϕ(λ)(n) = 0, ϕ(λ)(n+1) = 0
if and only if
ψ(eiθλ)(n) = 0, ψ(eiθλ)(n+1) = 0.
Take θ = 2πj/k, j = 1, . . . , k − 1. Then Gj, j = 1, . . . , k, are isometry each
other.
We should mention that, Corollary 5 can be deduced from [16, Theorem
2.4]. However, in view of the analysis, our idea is totally different from that of
[16].
4 Algebraic multiplicity for two-dimensional nonnegative tensors
In this section, we will establish the algebraic multiplicity for order m dimension
two tensors by the property of its resultant.
Suppose that A is a tensor, and
x = (x1, x2), y= (xm−1
1 , xm−2
1 x2, . . . , xm−1
2 ), B= (bij)2×m,
which satisfy
By = A xm−1.
Define
C = (cij)(2m−2)×(2m−2) ,
where
cij =
⎧⎪⎨
⎪⎩
b1k, 1 k m, 1 i m − 1, j = k + i − 1,
b2k, 1 k m, m i 2m − 2, j = k + i − m,
0, otherwise.
1142 Yiyong LI, et al.
Then we can get that the resultant of A is det(C − λE). Thus, the algebraic
multiplicity of A can be obtained by the property of C.
Lemma 4 Assume that A is an order m dimension two nonnegative
irreducible tensor and b1ji1
= 0, 1 i1 r1, 1 ji1 < m (resp. b2ki2
= 0,
1 i2 r2, 1 < ki2 m). Let (a, b) be the greatest common divisor of a and
b, q = (m− 1, j1 −1, . . . , jr1
− 1) (resp. q = (m−1, k1 − 1, . . . , kr2
−1)). Then
C can be divided into q same irreducible blocks.
Proof Since A is an order m dimension two nonnegative irreducible tensor, C
is a nonnegative matrix and b1m = 0, b21 = 0.
Case 1 r = 2r1, where r = 2m − 2, r1 = m − 1.
For b21 = 0,
patha11: 1 → r1 + 1 → 1,
patha12: 2 → r1 + 2 → 2,
. . . ,
patha1r1 : r1 → 2r1 → r1.
Case 2 For b1j1
= 0,
(1) r1 = k1r2 + r3, where r2 = j1 − 1,
patha21: 1 → r2 + 1→· · ·→k1r2 + 1
→ r1 + r2 − r3 + 1
−−−−−−−−−−→
patha1r1−r3+1r2 − r3 + 1,
. . . ,
patha2r3 : r3 → r2 + r3 →· · ·→k1r2 + r3 → r1 + r2
−−−−−→
patha1r2r2,
patha2r3+1 : r3 + 1 → r2 + r3 + 1→· · ·→k1r2 + r3 + 1
−−−−−→
patha111,
. . . ,
patha2r2 : r2 → 2r2 →· · ·→k1r2 + r2
−−−−−−−−−−→
patha1r2−r3+1r2 − r3;
(2) r2 = k2r3 + r4,
patha31: 1
−−−−−→
patha21r2 − r3 + 1
−−−−−−−−−→
pathar2−r3+1 · · ·
−−−−−−−−−−−→
patha2r2−k2r3+1r4 + 1,
. . . ,
patha3r3−r4 : r3 − r4
−−−−−−−−→
patha2r3−r4r2 − r4
−−−−−−−→
pathar2−r4
· · ·
−−−−−−−−−−−−−−→
patha2r2−(k2−2)r3−r4r3,
patha3r3−r4+1 : r3 − r4 + 1
−−−−−−−−−−→
patha2r3−r4+1r2 − r4
+1
−−−−−−−−−→
pathar2−r4+1 · · ·
−−−−−−−−→
patha2r3+1)1,
. . . ,
patha3r3 : r3
−−−−−→
patha2r3r2
−−−−−→
patha2r2r2 − r3
−−−−−−−−→
patha2r2−r3
· · ·
−−−−−−−−−−−−→
patha2r2−(k2−1)r3r4.
Continuing the process:
New definition of geometric multiplicity of eigenvalues of tensors 1143
(p) rp = kprp+1,
pathap+11: 1
−−−−−→
pathap1rp − rp+1 + 1· · ·
−−−−−−−−−−−−−→
pathaprp−kprp+1+11,
. . . ,
pathap+1rp+1 : rp+1
−−−−−−−→
patha2rp+1rp · · ·
−−−−−−−−−−−−−−→
patha2rp−(kp−1)rp+1rp+1,
or
pathap+11: 1
−−−−−→
pathap1rp+1 + 1· · ·
−−−−−−−−−−−−−→
pathap(kp−1)rp+1+1rp + 1
−−−−−−−→
pathaprp+11,
. . . ,
pathap+1rp+1 : rp+1
−−−−−−−→
pathaprp+1rp
−−−−−→
pathaprprp − rp+1 · · ·
−−−−−−−−−−−−−→
pathaprp(kp−2)rp+1rp+1.
In (1),
patha2k(m−1,j1−1)+i, 1 i (m − 1, j1 − 1), k(m − 1, j1 − 1) = j1 − 1,
have the same number and
ci1,j = ci2,j , 1 i1, i2 (m − 1, j1 − 1),
where ci,j is the j-th line segment of the patha2k(m−1,j1−1)+i. The same result
can be got for 0 k r2/(m − 1, j1 − 1). So let rp+1 = (m−1, j1−1), b1j1
= 0,
b21 = 0 make C have rp+1 = (m − 1, j1 − 1) same irreducible blocks.
Case 3 For b1j2
= 0, let rk = (m − 1, j2 − 1), b1j2
= 0,b21 = 0 in order to C
has rk = (m − 1, j2 − 1) same irreducible blocks.
In Case 2,
patha1: 1→· · ·→rp+1 + 1→· · ·→1,
patharp+1 : rp+1 →· · ·→2rp+1 →· · · →rp+1.
In Case 3,
patha1: 1→· · · →rk +1→· · · →1,
pathark : rk →· · ·→2rk →· · ·→rk.
Then let r = (rp+1, rk) = (m − 1, j1 − 1, j2 − 1), b1j1
= 0, b1j2
= 0, b21 = 0
make C have r same irreducible blocks.
Repeating the same procedure, let q = (m − 1, j1 − 1, . . . , jr1
− 1). Then C
can be divided into q same irreducible blocks.
Lemma 5 Assume that A is an order m dimension two nonnegative
irreducible tensor and b1ji1
= 0, 1 i1 r1, 1 ji1 < m, b2ki2
= 0,
1 i2 r2, 1 < ki2 m. Let (a, b) be the greatest common divisor of a
and b, q = (m−1, j1 −1, . . . , jr1
−1, k1 −1, . . . , kr2
−1). Then C can be divided
into q same irreducible blocks.
Proof The conclusion can be reached by the same proof of Lemma 4.
1144 Yiyong LI, et al.
Theorem 13 Assume that A is an order m dimension two nonnegative
irreducible tensor and b1ji1
= 0, 1 i1 r1 < m, b2ki2
= 0, 2 i2 r2.
Let (a, b) be the greatest common divisor of a and b, q = (m−1, j1−1, . . . , jr1
−
1, k1 −1, . . . , kr2
−1). Then its characteristic polynomial φ(t) := det(tI −A ) is
of the form
φ(t) = {tm[tk − ρk(A )][tk − δ1ρk(A )] · · · [tk − δrρk(A )]}q,
where |δ| < 1 for 1 < i r if r > 1, and m + rk = (2m − 2)/q.
Proof C can be divided into q same irreducible blocks by Lemma 5. And every
block have order (2m − 2)/q. So every block have characteristic polynomial
φ(t) = tm[tk − ρk(A )][tk − δ1ρk(A )] · · · [tk − δrρk(A )],
where |δi| < 1 for 1 < i r if r > 1, and m + rk = n. This can be deduced
from [17, Corollary 2.12]. Thus, A have characteristic polynomial φ(t) :=
det(tI − A ) with the desired form.
Theorem 14 Assume that A is an order m dimension two nonnegative
irreducible tensor and b1ji1
= 0, 1 i1 r1, b2ki2
= 0, 1 i2 r2. Let
(a, b) be the greatest common divisor of a and b, q = (j1 − 1, . . . , jr1
− 1,m −
k1, k2−k1, . . . , kr2
−k1). Then its characteristic polynomial φ(t) := det(tI −A )
is of the form
φ(t) = tm−1−jr1+k1{tm[tk − ρk(A)][tk − δ1ρk(A)] · · · [tk − δrρk(A)]}q,
where |δ| < 1 for 1 < i r if r > 1, and m + rk = (m − 1 + jr1
− k1)/q.
Proof The conclusion can be reached by the same proof for Theorem 13.
Theorem 15 If A is an order m dimension two nonnegative irreducible
tensor for every eigenvalue with modulus ρ(A ), then its complex geometric
multiplicity =algebraic multiplicity.
Proof Since A is a nonnegative irreducible tensor with dimension two, we
can suppose b1ji1
= 0, 1 i1 r1, b2ki2
= 0, 1 i2 r2, q = (j1 − 1,
. . . , jr1
− 1,m − k1, . . .,m − kr2). Assume that G is the set of eigenvectors
corresponding to ρ(A ). Let x = (x1, x2) ∈ A. Then
A xm−1 = ρ(A )x[m−1].
Thus,
n
j2,...,jm=1
aij2···jmxj2
· · · xjm = xm−1
i , i= 1, 2.
n
j2,...,jm=1
aij2···jm
|xj2
| · · · |xjm
| = |xi|m−1, i= 1, 2.
New definition of geometric multiplicity of eigenvalues of tensors 1145
n
j2,...,jm=1
aij2···jm
|xj2
| · · · |xjm
| =

n
j2,...,jm=1
aij2···jmxj2
· · · xjm

, i= 1, 2.
Hence, we can get
b1ji1
x
m−ji1
1 x
ji1
−1
2 =
x1
|x1|
m−1
|x1|m−ji1 |x2|ji1
−1, 1 i1 r1;
b2ki2
x
m−ki2
1 x
ki2
−1
2 =
x2
|x2|
m−1
|x1|m−ki2 |x2|ki2
−1, 1 i2 r2.
So we have x1
|x1|
ji1
−1
=
x2
|x2|
ji1
−1
, 1 i1 r1,
x1
|x1|
m−ki2 =
x2
|x2|
m−ki2 , 1 i2 r2.
Then
x[q] = eiθ|x|[q].
And if
x[q] = eiθ|x|[q], |x| ∈ A,
then x ∈ A. Since
q = (j1 − 1, . . . , jr1
− 1,m − k1, . . .,m − kr2)
= (j1 − 1, . . . , jr1
− 1,m − k1, k2 − k1, . . . , kr2
− k1),
its complex geometric multiplicity = algebraic multiplicity.
Acknowledgements The authors are grateful to Mr. Xi He and Mr. Zhongming Chen
for their helpful discussion. And the authors would like to thank the reviewers for their
suggestions to improve the presentation of the paper. This work was supported in part by
the National Natural Science Foundation of China (Grant No. 11271206) and the Natural
Science Foundation of Tianjin (Grant No. 12JCYBJC31200).
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