Trans. Tianjin Univ. 2014, 20: 232-235

DOI 10.1007/s12209-014-2110-4

Accepted date: 2013-03-01.

Wu Wei, born in 1969, female, Dr, associate Prof.

Correspondence to Wu Wei, E-mail: wuwei0423@126.com.

E-characteristic Polynomials of Real Rectangular Tensor

Wu Wei(吴 伟)，Chen Xiaoxiao(陈肖肖)

(School of Sciences, Tianjin University, Tianjin 300072, China)

© Tianjin University and Springer-Verlag Berlin Heidelberg 2014

Abstract：By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the E-characteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated. Keywords：E-characteristic polynomial; rectangular tensor; E-singular value; rank-one approximation

Qi[1] and Lim[2] independently proposed the con-cepts of eigenvalues of tensors in 2005, and since then the eigenvalue problems of tensors have attracted atten-tion in various fields. Applications of eigenvalues of the square tensors include medical resonance imaging[3], higher-order Markov chains[4], positive definiteness of even-order multivariate forms in automatic control[5], and the best rank-one approximation in data analysis[6]. The rectangular tensors arise from the strong ellip-ticity condition problem in solid mechanics and the en-tanglement problem in quantum physics. Chang et al de-fined the singular values of rectangular tensors and stud-ied their properties[7]. Recently, the E-singular values of a real partially symmetric rectangular tensor were defined, and some properties of the rectangular tensors such as positive definiteness and orthogonal similarity were in-vestigated[8]. The E-characteristic polynomials of the square ten-sors were introduced in Refs. [9]—[11]. However, little exploration on the E-characteristic polynomials of the rectangular tensors has been conducted.

In this paper, we define the E-characteristic poly-nomials of the rectangular tensors and discuss their prop-erties relative to the E-singular values. Moreover, we are interested in the best rank-one approximation of a real partially symmetric rectangular tensor.

1 Preliminaries

First we recall the basic definitions on square ten-

sors[1] and rectangular tensors[7]. Assume that , ,p q m and n are positive integers, , 2m n≥ , then

1 1( )

p qi i j jaA

where 1 1p qi i j ja R , for 1, ,ki m , 1, ,k p and

1, ,kj n , 1, ,k q , a real ( , )p q th order ( ) -m n di-mensional rectangular tensor, or simply a real rectangular tensor. Let M p q and m n N . In this case, we call A a real Mth order N-dimensional square tensor, or simply a real square tensor. When 2p q and

2m n (or 3), such a tensor is the elasticity tensor[12-14]. Two same-sized rectangular tensors are added entry-by-

entry. The rectangular tensor A is called partially symmet-

ric[7], if 1 1p qi i j ja is invariant under any permutation of

indices among 1, , pi i , and any permutation of indices among 1, , qj j , i.e.,

1 1( ) ( ) , ,p qi i j j p qa S S

where rS ( ,r p q ) is the permutation group of r indi-ces. When 2p q , such a partially symmetric rectangu-lar tensor is useful for the entanglement problem in quan-tum physics[15]. For a real m-vector T

1( , , )mx x x and a real n -

vector T1( , , )ny y y , p qx y denotes a real ( , )p q th or-

der ( )m n -dimensional rank-one rectangular tensor with the following entries:

1 1

( )p q

p qi i j jx x y yx y

where 1 1p qi i j jx x y y R , for 1, ,ki m , 1, ,k p

and 1, ,kj n , 1, ,k q . Let

Wu Wei et al: E-characteristic Polynomials of Real Rectangular Tensor

—233—

( , ) p qf x y Ax y

1 1 1 1

1 1, , 1 , , 1p q p q

p q

m n

i i j j i i j ji i j j

a x x y y

When both p and q are even, if ( , ) 0f x y for all mx R , x 0 , ny R , y 0 , then A is positive defi-

nite. Let 1p qAx y be a vector in mR such that

1( )p qi

Ax y

2 1 2 1

2 1, , 1 , , 1

, 1, ,p q p q

p q

m n

i i j j i i j ji i j j

a x x y y i m

Similarly, let 1p qAx y be a vector in nR such that 1( )p q

j Ax y

1 2 1 2

1 2, , 1 , , 1

, 1, ,p q p q

p q

m n

i i j j i i j ji i j j

a x x y y j n

With the definition of E-singular values given in Ref. [8], considering

1

1

T

T

1

1

p q

p q

Ax y x

Ax y y

x x

y y

(1)

If C , {\ }mx C 0 and {\ }ny C 0 are solutions of Eq.(1), then we call an E-singular value of A , and x and y the left and right E-singular vectors of A , associ-ated with the E-singular value . If R , mx R and

ny R are solutions of Eq.(1), then we call a Z-

singular value of A , and x and y the left and right Z-

singular vectors of A , associated with the Z-singular value .

2 E-characteristic polynomial of real rec-tangular tensor

Throughout this section, unless stated otherwise, A refers to a real ( , )p q th order ( )-m n dimensional rectan-gular tensor and it is not necessarily partially symmetric; in other words, A is generic. Let L m n .

For the M-eigenvalue of a 4-order real partially symmetric tensor, Theorem 1 and Theorem 4 in Ref. [12] proved that the M-eigenvalues always exist, and if two 4-

order partially symmetric tensors are orthogonally simi-lar, then they have the same M-eigenvalues. These prop-erties have been extended to the E-singular values of a real partially symmetric rectangular tensor in Theorem 1(a) and Theorem 3 in Ref. [8].

The M-characteristic polynomial of a 4-order par-tially symmetric tensor was defined in Ref. [12]. For a

real square tensor (not necessarily symmetric), the E-

characteristic polynomial of such a square tensor was defined in Ref. [9]. However, there is no generalization of the E-characteristic polynomial for a real rectangular tensor at present.

We now divide the definitions of the E-characteristic polynomial into the following cases.

Case 1 Suppose that both p and q are even. Let

11 T T2 2

11 T T2 2

( ) ( )( , )

( ) ( )

p qp q

p qp q

F

Ax y x x y y xx y

Ax y x x y y y

(2)

According to the resultant theory in Ref. [16], the resultant of ( , )F x y is a polynomial of the coefficient of

( , )F x y , i.e., a 1-dimensional polynomial of , which vanishes as long as ( , )F x y 0 has a nonzero solution ( , ) ( , )x y 0 0 in LC . We call this polynomial of as the E-characteristic polynomial of A , denoted by ( ) . When 2p q , ( ) is the M-characteristic polynomial of A defined in Ref. [12].

Case 2 Suppose that one of p and q is odd and the other is even. Without loss of generality, we may suppose that p is odd, 3p≥ , and q is even. Let

1 2 T 20

10 1 T 20

2 T0

( )

( , , )( )

qp q p

qp q p

x

G xx

x

Ax y y y x

x yAx y y y y

x x

(3)

According to the resultant theory in Ref. [16], the resultant of 0( , , )G x x y is a polynomial of the coefficient of 0( , , )G x x y , i.e., a 1-dimensional polynomial of , which vanishes as long as 0( , , )G x x y 0 has a nonzero solution ( , ) ( , )x y 0 0 in LC . We call this polynomial of as the E-characteristic polynomial of A , denoted by

( ) . Case 3 Suppose that both p and q are odd, and

, 3p q≥ . Let

1 20 0

1 20 0

0 0 2 T0

2 T0

( , , , )

p q p q

p q p q

x y

x yH x y

x

y

Ax y x

Ax y yx y

x x

y y

(4)

Again, according to the resultant theory in Ref. [16], the resultant of 0 0( , , , )H x y x y is a polynomial of the co-efficient of 0 0( , , , )H x y x y , i.e., a 1-dimensional polyno-mial of , which vanishes as long as 0 0( , , , )H x y x y 0 has a nonzero solution ( , ) ( , )x y 0 0 in LC . We call this polynomial of as the E-characteristic polynomial of A , denoted by ( ) .

Transactions of Tianjin University Vol.20 No.3 2014

—234—

According to Ref. [7], 1{ }mie and 1{ }n

jf denote the canonical basis of mR and nR , respectively. Let

pi i i e e e ( p times) and q

j j j f f f ( q times), where is the notation of tensor product of vec-tors.

For any 1, ,j n , let 1 ,( , ) ( )

p

qj i i j ja A f be a pth

order m-dimensional square tensor. For any 1, ,i m , let

1,( , ) ( )q

pi i i j ja A e be a qth

order n-dimensional square tensor. For a real Mth order N-dimensional square tensor

A , let 1M Ax be a vector in NR such that

2 2

2

1

, , 1

( ) , 1, ,M M

M

NM

i i i i ii i

a x x i N

Ax

Definition 1 (Regularity of square tensors in Ref. [12])

A real Mth order N-dimensional square tensor A is called irregular if there is an Nx C such that x 0 ,

1M Ax 0 and T 0x x . Otherwise, A is regular. Definition 2 (Regularity of rectangular tensors) A real rectangular tensor A of ( , )p q th order

( )m n -dimensional is called regular if all the square ten-sors ( , )q

jA f ( 1, , )j n and ( , )pi A e ( 1, , )i m are

regular in the sense of Definition 1. Theorem 1 Suppose that , 3p q≥ , then an E-

singular value of a real ( , )thp q order ( )-m n dimen-sional rectangular tensor is always a root of the E-

characteristic polynomial of the rectangular tensor. Con-versely, even if the rectangular tensor is regular, the con-clusion may not be true.

Proof Suppose that A is a real ( , )thp q order ( )-m n dimensional rectangular tensor and , 3p q≥ . Let be an E-singular value of A , and x and y be the left and right E-singular vectors of A , associated with . By Eq.(2), when both p and q are even, ( , )x y is a non-zero solution of ( , )F x y 0 for that ; when one of p and q is odd and the other is even, for Case 2, we may suppose that p is odd and q is even, and

T0( , , )x x x x y is a nonzero solution of 0( , , )G x x y 0

for that ; when both p and q are odd, T T

0 0( , , , )x y x x y y x y is a nonzero solution of

0 0( , , , )H x y x y 0 for that . According to the resultant theory[16] and our definition, we have ( ) 0 .

On the other hand, suppose that A is regular. Let be a root of ( ) 0 . By the resultant theory[16] and our definition, ( , )F x y 0 , 0( , , )G x x y 0 and 0 0( , , , )H x y x y 0 have a nonzero solution ( , ) ( , )x y 0 0 in LC . If there exists an mx C , x 0 or a ny C , y 0 for that , the conclusion is surely untrue, since all the E-singular

vectors are nonzero. Even if both x and y are two non-zero vectors, it is difficult to verify the contradiction with regularity.

It is easy to see that the generalized E-characteristic polynomials of rectangular tensors are more complicated than the E-characteristic polynomials of square tensors. However, if we fix the variable y and let y 0 in Eqs.(2), (3) and (4), then they can be regarded as the polynomial systems of variable x . Now, the E-

characteristic polynomial is denoted by ( ) y , and y is regarded as an n-vector of indeterminate variables { | 1, , }jy j n in the remainder of this section. Simi-larly, if we fix variable x and let x 0 , then we can get the same argument. For convenience, we only discuss fixing variable y here. Especially, let jy f and

( ) ( )j

y f , for any 1, ,j n , then we have the fol-lowing weak result of the opposite of Theorem 1. Fur-thermore, ( )j jy y f is used to denote both a vector of indeterminate variables { | 1, , }jy j n and a specific vector in nR , which are clear from the content.

Theorem 2 Suppose that , 3p q≥ and A is a real ( , )thp q order ( )m n -dimensional rectangular tensor. If A is regular, then a root of the E-characteristic polyno-

mial ( )j

f is an E-singular value of A , for any 1, ,j n . Proof Suppose that A is regular. Let be a root of

( )j

f , for any 1, ,j n . By the resultant theory[16] and our definition, when both p and q are even, there is an

{\ }mx C 0 such that ( , )jF x f 0 ; for Case 2, when p is odd and q is even, there is an {\ }mx C 0 and an

0x C such that 0( , , )jG x x f 0 ; when both p and q are odd, there is an {\ }mx C 0 , an 0x C and a 0y C such that 0 0( , , , )H x y x y 0 . If T 0x x , then by

( , )jF x f 0 , 0( , , )jG x x f 0 or 0 0( , , , )H x y x y 0 , we have 1p q

j Ax f 0 . As the jth component of jf is one

and zeros elsewhere, we get 1( , )p qj

A x f 0 . Conse-quently, the square tensor ( , )q

jA f is irregular, for any 1, ,j n . This contradicts with the assumption that A

is regular. Hence, T 0x x . Let

T

x

xx x

We see that Eq.(1) is satisfied with , x and jf . This implies that is an E-singular value of A .

3 Best rank-one approximation

Suppose that A is a real ( , )p q th order ( )m n -

dimensional partially symmetric rectangular tensor, and

Wu Wei et al: E-characteristic Polynomials of Real Rectangular Tensor

—235—

both p and q are even throughout this section. The Frobenius norm of A is defined as

1 1

1 1

2 1/ 2

, , 1 , , 1

|| || ( )( )p q

p q

m n

F i i j ji i j j

a

A

For R , mx R and ny R , p qx y is a rank-one ( , )thp q order -dimensi n l) a( om n partially symmetric rectangular tensor with elements

1 1p qi i j jx x y y . We say that p qx y is the best rank-one approximation of A , if R , mx R , T 1x x and ny R , T 1y y mini-mize || ||p q

FA x y . Lemma 1 (Theorem 1(a) and Theorem 2 in Ref. [8]) Z-singular values always exist. If x and y are the

left and right Z-singular vectors of A associated with a Z-singular value , then p q Ax y Moreover, A is positive definite if and only if the small-est Z-singular value of the real partially symmetric rec-tangular tensor A is positive.

Theorem 3 If is the Z-singular value of A with the largest absolute value, and x and y are the corre-sponding left and right Z-singular vectors, then p qx y is the best rank-one approximation of A .

Proof Let T 1x x and T 1y y . We have 2 2=|| || || 2p q p q

F F x y A Ax yA

2 T T( ) ( )p q x x y y

2 2|| || 2 p qF A Ax y

Its minimum is attained when p q Ax y . Hence, it holds that 2 T Tmin{|| || : , 1, 1}p q

F RA x y x x y y

2 2 T Tmin{|| || ( ) : 1, 1}p qF A Ax y x x y y

2 2 T T|| || max{( ) : 1, 1}p qF A Ax y x x y y

By Lemma 1 and the assumption, the conclusion follows.

4 Conclusions

The E-characteristic polynomial of a real rectangular tensor is proposed to investigate E-singular values. The connection between the E-singular values and the best rank-one approximation of a partially symmetric tensor is established.

References

［1］ Qi L Q. Eigenvalues of a real supersymmetric tensor[J].

Journal of Symbolic Computation, 2005, 40(6): 1302-

1324.

［2］ Lim L H. Singular values and eigenvalues of tensors: A

variational approach[C]. In: Proceedings of the IEEE In-

ternational Workshop on Computational Advances in

Multi-Sensor Adaptive Processing. Puerto Vallarta, Mexico,

2005.

［3］ Qi L Q, Wang Y J, Wu E X. D-eigenvalues of diffusion

kurtosis tensors[J]. Journal of Computational and Applied

Mathematics, 2008, 221(1): 150-157.

［4］ Ng M, Qi L Q, Zhou G L. Finding the largest eigenvalue of

a nonnegative tensor[J]. SIAM Journal on Matrix Analysis

and Applications, 2009, 31(3): 1090-1099.

［5］ Qi L Q, Wang F, Wang Y J. Z-eigenvalue methods for a

global polynomial optimization problem[J]. Mathematical

Programming, 2009, 118(2): 301-316.

［6］ De Lathauwer L, De Moor B, Vandewalle J. On the best

rank-1 and rank- 1 2( , , , )NR R R approximation of higher-

order tensors[J]. SIAM Journal on Matrix Analysis and

Applications, 2000, 21(4): 1324-1342.

［7］ Chang K, Qi L Q, Zhou G L. Singular values of a real rec-

tangular tensor[J]. Journal of Mathematical Analysis and

Applications, 2010, 370(1): 284-294.

［8］ Zhao N. E-singular values of a real partially symmetric

rectangular tensor[J]. Journal of Shandong Univer-

sity(Natural Science), 2012, 47(10):34-37 (in Chinese).

［9］ Qi L Q. Eigenvalues and invariants of tensors[J]. Journal

of Mathematical Analysis and Applications, 2007,

325(2):1363-1377.

［10］ Li A M, Qi L Q, Zhang B. E-characteristic polynomials of

tensors[J]. Communications in Mathematical Sciences,

2013, 11(1): 33-53.

［11］ Hu S L, Qi L Q. The E-characteristic polynomial of a ten-

sor of dimension 2[J]. Applied Mathematics Letters, 2013,

26(2): 225-231.

［12］ Qi L Q, Dai H H, Han D R. Conditions for strong elliptic-

ity and M-eigenvalues[J]. Frontiers of Mathematics in

China, 2009, 4(2): 349-364.

［13］ Ling C, Nie J W, Qi L Q et al. Biquadratic optimization

over unit spheres and semidefinite programming relaxa-

tions[J]. SIAM Journal on Optimization, 2009, 20(3):

1286-1310.

［14］ Dahl G, Leinaas J M, Myrheim J et al. A tensor product

matrix approximation problem in quantum physics[J].

Linear Algebra and Its Applications, 2007, 420(2/3): 711-

725.

［15］ Wang Y J, Qi L Q, Zhang X Z. A practical method for

computing the largest M-eigenvalue of a fourth-order par-

tially symmetric tensor[J]. Numerical Linear Algebra with

Applications, 2009, 16(7): 589-601.

［16］ Cox D A, Little J, O'Shea D. Using Algebraic Geome-

try[M]. Springer-Verlag, New York, USA, 1998.

(Editor: Wu Liyou)