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Some inequalities on the minimum H-eigenvalue of the Fan product of Z -tensors
Linear Algebra and its Applications 579 (2019) 55–71
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Some inequalities on the minimum H-eigenvalue of the Fan product of Z-tensors Jihong Shen a,b , Yue Wang a , Changjiang Bu a,b,∗ a
College of Automation, Harbin Engineering University, Harbin 150001, PR China College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, PR China b
a r t i c l e
i n f o
Article history: Received 23 November 2018 Accepted 24 May 2019 Available online 29 May 2019 Submitted by J. Shao MSC: 15A69 15A18 65F15
a b s t r a c t In this paper, some bounds on the minimum H-eigenvalue of a Z-tensor and the Fan product of Z-tensors are given. By applying the bounds, we obtain some criteria for M-tensors and the positive definiteness of the Fan product of Z-tensors, and give some properties for the determinant of the Fan product of M-tensors. © 2019 Elsevier Inc. All rights reserved.
Keywords: Z-tensor H-eigenvalue Fan product Positive definiteness
1. Introduction For a positive integer n, let [n] = {1, 2, . . . , n}. An m-order n-dimensional tensor A = (ai1 i2 ···im ) (ij ∈ [n] , j ∈ [m]) is a multidimensional array with nm entries. We sometimes write ai1 i2 ···im as Ai1 i2 ···im . When m = 2, A is an n × n matrix. The tensor A is called nonnegative if all its entries are nonnegative, denoted by A ≥ 0. A tensor whose diagonal * Corresponding author. E-mail address: [email protected] (C. Bu). https://doi.org/10.1016/j.laa.2019.05.031 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
entries are ones and off-diagonal entries are zeros is called the unit tensor, denoted by I. Let C [m,n] and R[m,n] be the sets of m-order n-dimensional complex tensors and real tensors, respectively. Denote C n (resp. Rn ) the set of n-dimensional vectors over C (resp. R). In 2005, the concept of eigenvalues of tensors was proposed by Qi [19] and Lim [17], independently. For a tensor A = (ai1 i2 ···im ) ∈ C [m,n] , λ ∈ C is called an eigenvalue of A T if there exists a nonzero vector x = (x1 , . . . , xn ) ∈ C n such that Axm−1 = λx[m−1] ,
(1.1)
where Axm−1 is a vector in C n whose i-th component is (Axm−1 )i =
n
aii2 ···im xi2 · · · xim (i = 1, . . . , n),
i2 ,...,im =1
T and x[m−1] = xm−1 , ..., xm−1 . If λ ∈ R and x ∈ Rn , then λ is called an Hn 1 eigenvalue of A. Let σ(A) denote the set of all the eigenvalues of A. Denote ρ(A) = max {|λ| : λ ∈ σ(A)} the spectral radius of A and τ (A) = min {Re(λ) : λ ∈ σ(A)}. A tensor A = (ai1 i2 ···im ) is called symmetric if entries ai1 i2 ···im are invariant under any permutation of indices i1 · · · im . An m-th degree homogeneous polynomial f (x) can be represented by a symmetric tensor A as f (x) = Axm :=
n
ai1 i2 ···im xi1 xi2 · · · xim .
i1 ,i2 ,...,im =1
A symmetric tensor A = (ai1 i2 ···im ) ∈ R[m,n] is called positive definite (resp. positive semidefinite) if f (x) > 0 (resp. ≥ 0) for all 0 = x ∈ Rn . When A is a positive definite tensor, m must be even [19]. The positive definiteness of a homogeneous polynomial plays an important role in the stability study of nonlinear systems via Lyapunov’s direct method [18], such as tests for filters [1,3], the multivariate network realizability theory [4], and the output feedback stabilization problems [2]. In [19], Qi proved that a real symmetric tensor A is positive definite if and only if the minimum H-eigenvalue of A is positive. A tensor A ∈ R[m,n] is called a Z-tensor if all of its off-diagonal entries are nonpositive. A Z-tensor A = sI −B is called an M-tensor (resp. a nonsingular M-tensor) if s ≥ ρ(B) (resp. s > ρ(B)), where s ∈ R and B ≥ 0. Clearly, M-tensors are Z-tensors. Zhang et al. proved that a symmetric Z-tensor is a nonsingular M-tensor if and only if it is positive definite [26]. Gowda et al. showed that τ (A) is the minimum H-eigenvalue of a Z-tensor A (see [10]). A Z-tensor A is a nonsingular M-tensor if and only if τ (A) > 0 (see [26]). Some bounds on the minimum H-eigenvalue of Z-tensors and M-tensors were given [12, 22]. More results on M-tensors can be found in [6,20,21,28].
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For A = (aij ), B = (bij ) ∈ R[2,n] , the Fan product A B is a matrix with entries (A B)ij = (−1)δij +1 aij bij , where δij = 1 if i = j, and δij = 0 otherwise. In 1964, this product was proposed by Ky Fan and it was proved that this product of M -matrices is also an M -matrix [7]. Some bounds on τ (A B) for M -matrices A and B were given by τ (A), τ (B) or the entries of A and B [5,8,13]. It is natural to study the Fan product of tensors. For A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] , the Fan product A B is a tensor with entries (A B)i1 i2 ···im = (−1)δi1 i2 ···im +1 ai1 i2 ···im bi1 i2 ···im , where δi1 i2 ···im = 1 if i1 = i2 = · · · = im , and δi1 i2 ···im = 0 otherwise [15]. Clearly, the Fan product of Z-tensors is a Z-tensor. In this paper, we give some bounds on the minimum H-eigenvalue of a Z-tensor. And some bounds on the minimum H-eigenvalue τ (A B) for Z-tensors A and B are established with τ (A) and τ (B). Further, we use the bounds to give some criteria to determine M-tensors and the positive definiteness of the Fan product of Z-tensors. And some properties for the determinant of the Fan product of M-tensors are obtained. 2. Preliminaries In this section, we introduce some lemmas which are used to obtain the main results in Sections 3 and 4. Lemma 2.1. [19] Let A = (ai1 i2 ···im ) ∈ C [m,n] . Then σ(A) ⊆
n
{z ∈ C : |z − ai···i | ≤ Ri (A)},
i=1
where Ri (A) =
|aii2 ···im |.
(i2 ,...,im )=(i,...,i)
Denote Rn++ the set of n-dimensional positive vectors. The following lemma is Lemma 4.1 in [22]. Lemma 2.2. [22] Let A = (ai1 ···im ) ∈ R[m,n] be a Z-tensor. Then (Axm−1 )i ≤ τ (A) ≤ min aii···i 1≤i≤n 1≤i≤n xm−1 i min
T
for any x = (x1 , . . . , xn ) ∈ Rn++ . Using similar proof of the above lemma given in [22], we have Inequality (2.2).
(2.1)
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Lemma 2.3. Let A = (ai1 ···im ) ∈ R[m,n] be a Z-tensor. Then (Axm−1 )i 1≤i≤n xm−1 i
τ (A) ≤ max
(2.2)
T
for any x = (x1 , . . . , xn ) ∈ Rn++ . T
Proof. Let A = sI − B, where s ∈ R and B ≥ 0. For any x = (x1 , . . . , xn ) ∈ Rn++ , we obtain sxm−1 (Bxm−1 )i (Axm−1 )i − (Bxm−1 )i i = = s − . xm−1 xm−1 xm−1 i i i By the definition of eigenvalues of tensors, we know that τ (A) = s − ρ(B) (see [10]). Lemma 5.3 in [24] shows that min 1≤i≤n (Axm−1 )i s − min s − xm−1 . Hence 1≤i≤n
Bxm−1 xim−1
i
≤ ρ(B) ≤ max
1≤i≤n
Bxm−1 xim−1
i
. Then s − ρ(B) ≤
i
(Axm−1 )i . 1≤i≤n xm−1 i
τ (A) ≤ max
2
For A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] , A ≤ B means that ai1 ···im ≤ bi1 ···im for all ij ∈ [n] , j ∈ [m]. Lemma 2.4. Let {Ak } be a sequence of Z-tensors satisfying Ak ≤ Ak+1 for each positive integer k, and lim Ak = A. Then k→∞
lim τ (Ak ) = τ (A).
k→∞
Proof. We know that there exist s ∈ R, Bk ≥ 0 and B ≥ 0 such that Ak = sI − Bk and A = sI − B. So τ (Ak ) = s − ρ(Bk ). Since Ak ≤ Ak+1 and lim Ak = A, we have k→∞
Bk+1 ≤ Bk and lim Bk = B. Proposition 3.1 in [22] gives that lim ρ(Bk ) = ρ(B). Then k→∞
k→∞
lim τ (Ak ) = lim (s − ρ(Bk )) = s − lim ρ(Bk ) = s − ρ(B) = τ (A).
k→∞
k→∞
k→∞
2
For a Z-tensor A, Lemma 2.5 is a criterion to determine that A is an (resp. a nonsingular) M-tensor. Lemma 2.5. [26] Let A ∈ R[m,n] be a Z-tensor. Then (1) A is an M-tensor if and only if τ (A) ≥ 0; (2) A is a nonsingular M-tensor if and only if τ (A) > 0.
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Lemma 2.6. [12] Let A and B be Z-tensors satisfying A ≤ B. If A is an M-tensor, then B is an M-tensor and τ (A) ≤ τ (B). The following inequality is a transformation of Hölder’s Inequality, which is used to prove Theorem 3.10. Lemma 2.7. [11] If pj > 0 (j ∈ [s]) and
s j=1
where xji ≥ 0, i ∈ [n].
1 pj
≥ 1, then
s n
x ji ≤
i=1 j=1
s j=1
n i=1
p xjij
p1
j
,
3. Bounds for the minimum H-eigenvalue of Z-tensors In this section, we give some bounds on the minimum H-eigenvalue of a Z-tensor and the Fan product of Z-tensors. The lower bounds can be used to determine Z-tensors are (nonsingular) M-tensors and the positive definiteness of Z-tensors. For a set α, |α| denotes the cardinality of α. Let A = (ai1 ···im ) ∈ C [m,n] and α ⊂ [n] with |α| = r. A principal subtensor of A with index set α is A[α] = (ai1 ···im ), where i1 , . . . , im ∈ α (see [16]). In [14], Hu et al. proved that ρ(A[α]) ≤ ρ(A). For a Z-tensor A = sI − B, we know that τ (A) = s − ρ(B), where s ∈ R and B ≥ 0. Hence, τ (A[α]) = s − ρ(B[α]), where B[α] is a principal subtensor of B. Since ρ(B[α]) ≤ ρ(B), we obtain the following theorem. Theorem 3.1. Let A ∈ R[m,n] be a Z-tensor. Then τ (A) ≤ τ (A[α]), where α ⊂ [n]. Let Rmin (A) = min
n
1≤i≤n i2 ,...,im =1
aii2 ···im and Rmax (A) = max
n
1≤i≤n i2 ,...,im =1
aii2 ···im be the
smallest and the largest row sums of a tensor A = (ai1 i2 ···im ) ∈ R[m,n] , respectively. A tensor A is called weakly reducible, if there exists some nonempty proper subset I ⊂ [n] such that ai1 i2 ···im = 0 for any i1 ∈ I and at least an ij ∈ [n]\I, j = 2, . . . , m. Otherwise, A is called weakly irreducible [9,25]. If A is nonnegative weakly irreducible, then ρ(A) is an eigenvalue of A and there exists a positive eigenvector corresponding to ρ(A) (see [25]). For an irreducible M-tensor A, we know that Rmin (A) ≤ τ (A) ≤ Rmax (A) (see [12]). We extend this result to general Z-tensors. Theorem 3.2. Let A = (ai1 i2 ···im ) ∈ R[m,n] be a Z-tensor. Then Rmin (A) ≤ τ (A) ≤ Rmax (A) .
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Proof. A Z-tensor A can be written as A = sI − B, where s ∈ R and B ≥ 0. Then Rmin (A) = s −Rmax (B) and Rmax (A) = s −Rmin (B). Since Rmin (B) ≤ ρ (B) ≤ Rmax (B) (see Lemma 5.2 of [24]), we have Rmin (A) ≤ τ (A) ≤ Rmax (A). 2 For M -matrices A = (aij ), B = (bij ) ∈ R[2,n] , Horn and Johnson obtained that τ (A B) ≥ τ (A)τ (B) (see [13]). Fang proved that τ (A B) ≥ min {aii τ (B) + bii τ (A)}− 1≤i≤n
τ (A)τ (B) ≥ τ (A)τ (B) (see [8]). We extend these results to tensors. Theorem 3.3. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be two tensors. (1) If A and B are Z-tensors, then τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B).
(3.1)
1≤i≤n
(2) If A and B are M-tensors, then τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B) ≥ τ (A)τ (B). 1≤i≤n
Proof. (1) We divide this proof into two cases. Case 1. A B is weakly irreducible. By the definition of the Fan product of tensors, we know that A and B are weakly irreducible. A Z-tensor A can be written as A = tI −D and τ (A) = t −ρ(D), where t ∈ R and D ≥ 0. Since D is nonnegative weakly irreducible, there exists a positive eigenvector corresponding to ρ(D). And this vector is also a positive eigenvector corresponding to τ (A). Then there exists x = (x1 , x2 , . . . , xn )T ∈ Rn++ such that Axm−1 = τ (A)x[m−1] . Similarly, there exists y = (y1 , y2 , . . . , yn )T ∈ Rn++ such that By m−1 = τ (B)y [m−1] . Hence (τ (A) − aii···i ) xm−1 = i
aii2 ···im xi2 · · · xim (i = 1, 2, . . . , n)
(i2 ,...,im )=(i,...,i)
and (τ (B) − bii···i ) yim−1 =
bii2 ···im yi2 · · · yim (i = 1, 2, . . . , n) .
(i2 ,...,im )=(i,...,i)
Let z = (z1 , z2 , . . . , zn )T ∈ Rn++ , where zi = xi yi for all i ∈ [n]. Then (A B) z m−1 i zim−1 = aii···i bii···i −
⎛ 1 zim−1
⎝
(i2 ,...,im )=(i,...,i)
⎞ |aii2 ···im | |bii2 ···im | xi2 yi2 · · · xim yim ⎠
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
⎛ ≥ aii···i bii···i − ⎛ ×⎝
1 zim−1
⎝
61
⎞
|aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎞
|bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i)
= aii···i bii···i − (aii···i − τ (A)) (bii···i − τ (B)) = aii···i τ (B) + bii···i τ (A) − τ (A)τ (B). By Lemma 2.2, we have τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B).
(3.2)
1≤i≤n
Case 2. A B is weakly reducible. Let Ak = A + k1 T and Bk = B + k1 T , where k is a positive integer and T = (ti1 i2 ···im ) is a tensor whose diagonal entries are zeros and off-diagonal entries are negative. Then Ak and Bk are weakly irreducible Z-tensors. And Ak and Bk are monotone increasing sequences with lim Ak = A and lim Bk = B, respectively. By Lemma 2.4, we obtain k→∞
k→∞
lim τ (Ak ) = τ (A) and lim τ (Bk ) = τ (B). Substituting Ak and Bk into Inequality
k→∞
k→∞
(3.2), and let k → ∞, Inequality (3.1) holds. (2) By Statement (1), we get τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)}−τ (A)τ (B). Since 1≤i≤n
A and B are M-tensors, by Lemma 2.2 and Lemma 2.5(1), we have τ (A) ≤ min aii···i , 1≤i≤n
τ (B) ≤ min bii···i , τ (A) ≥ 0, and τ (B) ≥ 0. Then we obtain the inequality of (2). 2 1≤i≤n
Next we show that M-tensors (resp. nonsingular M-tensors) are closed under Fan product, which can be seen in [23]. Theorem 3.4. Let A = (ai1 ···im ), B = (bi1 ···im ) ∈ R[m,n] be M-tensors (resp. nonsingular M-tensors). Then A B is an M-tensor (resp. a nonsingular M-tensor). Proof. By Theorem 3.3(2) and Lemma 2.5, we have τ (A B) ≥ τ (A)τ (B) ≥ 0 (resp. > 0). Then A B is an M-tensor (resp. a nonsingular M-tensor). 2 By Theorem 3.1, Theorem 3.3(2) and Lemma 2.5, we can get the following result immediately. Theorem 3.5. (Theorem 4.1 in [16]) Let A = (ai1 ···im ), B = (bi1 ···im ) ∈ R[m,n] be M-tensors (resp. nonsingular M-tensors). Then all principal subtensors of A B are M-tensors (resp. nonsingular M-tensors).
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For A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] , let αi (A) = βi (A) =
min
(i2 ,...,im )=(i,...,i)
|aii2 ···im |, and
max
(i2 ,...,im )=(i,...,i)
|aii2 ···im |,
Γ1 (A, B) = min {(aii···i − αi (A)) bii···i + αi (A)τ (B)} , 1≤i≤n
Γ2 (A, B) = max {(aii···i − βi (A)) bii···i + βi (A)τ (B)} , 1≤i≤n
1 1 1 1 Γ3 (A, B) = min aii···i bii···i − (αi (A)) 2 (αi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 , 1≤i≤n
Γ4 (A, B) = max
1≤i≤n
1 1 1 1 aii···i bii···i − (βi (A)) 2 (βi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 .
Here, we give new lower and upper bounds of τ (A B). Theorem 3.6. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then (1) Γ1 (A, B) ≤ τ (A B) ≤ Γ2 (A, B),
(3.3)
Γ1 (B, A) ≤ τ (A B) ≤ Γ2 (B, A),
(3.4)
max {Γ1 (A, B), Γ1 (B, A)} ≤ τ (A B) ≤ min {Γ2 (A, B), Γ2 (B, A)} ,
(3.5)
Γ3 (A, B) ≤ τ (A B) ≤ Γ4 (A, B).
(3.6)
(2)
(3)
(4)
Proof. (1) We prove the result under two cases. Case 1. B is weakly irreducible. Similarly to the proof of Theorem 3.3(1), there exists u = (u1 , u2 , . . . , un )T ∈ Rn++ such that Bum−1 = τ (B)u[m−1] . Then (τ (B) − bii···i ) um−1 = i
bii2 ···im ui2 · · · uim , i ∈ [n] .
(i2 ,...,im )=(i,...,i)
Hence (A B) um−1 i um−1 i
− aii···i bii···i um−1 i =
|aii2 ···im | |bii2 ···im | ui2 · · · uim
(i2 ,...,im )=(i,...,i) um−1 i
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
⎛ αi (A) ⎝ ≥ aii···i bii···i − m−1 ui (i
63
⎞
|bii2 ···im | ui2 · · · uim ⎠
2 ,...,im )=(i,...,i)
= aii···i bii···i − αi (A) (bii···i − τ (B)) = (aii···i − αi (A)) bii···i + αi (A)τ (B) and
(A B) um−1
i
um−1 i
⎛ βi (A) ⎝ ≤ aii···i bii···i − m−1 ui (i
⎞ |bii2 ···im | ui2 · · · uim ⎠
2 ,...,im )=(i,...,i)
= aii···i bii···i − βi (A) (bii···i − τ (B)) = (aii···i − βi (A)) bii···i + βi (A)τ (B). By Inequalities (2.1) and (2.2), we have Γ1 (A, B) ≤ τ (A B) ≤ Γ2 (A, B). Case 2. B is weakly reducible. By the proof of Case 2 in Theorem 3.3(1), we have Inequality (3.3). As the Fan product has the commutative law, Inequality (3.4) holds. From Inequalities (3.3) and (3.4), we obtain Inequality (3.5). (4) We divide this proof into two cases. Case 1. A B is weakly irreducible. Since A and B are weakly irreducible, similarly to the proof of Theorem 3.3(1), there exist x = (x1 , x2 , . . . , xn )T ∈ Rn++ and y = (y1 , y2 , . . . , yn )T ∈ Rn++ such that Axm−1 = τ (A)x[m−1] and By m−1 = τ (B)y [m−1] . Then
(τ (A) − aii···i ) xm−1 = i
aii2 ···im xi2 · · · xim ,
(i2 ,...,im )=(i,...,i)
(τ (B) − bii···i ) yim−1 =
bii2 ···im yi2 · · · yim .
(i2 ,...,im )=(i,...,i)
By Cauchy-Schwarz Inequality, we have
1
1
1
1
|aii2 ···im | |bii2 ···im | xi22 yi22 · · · xi2m yi2m
(i2 ,...,im )=(i,...,i)
⎛ ≤ ⎝
(i2 ,...,im )=(i,...,i)
⎞ 12 ⎛ a2ii2 ···im xi2 · · · xim ⎠ ⎝
(i2 ,...,im )=(i,...,i)
⎞ 12 b2ii2 ···im yi2 · · · yim ⎠
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⎛
⎞ 12
≤⎝
αi (A) |aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
×⎝
αi (B) |bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
= (αi (A)) (αi (B)) ⎝ 1 2
1 2
|aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
×⎝
|bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i) 1
m−1 2
1
1
= (αi (A)) 2 (αi (B)) 2 (aii···i − τ (A)) 2 xi
1
m−1 2
(bii···i − τ (B)) 2 yi
1
.
1
Let z = (z1 , z2 , . . . , zn )T ∈ Rn++ , where zi = xi2 yi2 for all i ∈ [n]. Hence (A B) z m−1 i zim−1
⎛
= aii···i bii···i −
1 zim−1
⎞
⎝
1 2 i2
1 2 i2
1 2 im
1 2 im
|aii2 ···im | |bii2 ···im | x y · · · x y ⎠
(i2 ,...,im )=(i,...,i) 1 2
1
1
1
≥ aii···i bii···i − (αi (A)) (αi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 and (A B) z m−1 i zim−1 ≤ aii···i bii···i − ⎛ ×⎝
⎛ 1 zim−1
(βi (A)) (βi (B)) ⎝
1 2
1 2
⎞ 12 |aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎞ 12 |bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i) 1
1
1
1
= aii···i bii···i − (βi (A)) 2 (βi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 . From Inequalities (2.1) and (2.2), we obtain Inequality (3.6). Case 2. A B is weakly reducible. By the proof of Case 2 in Theorem 3.3(1), we get Inequality (3.6). 2
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Next, we give a lower bound of τ (A B) with the entries of tensors A and B. Theorem 3.7. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then τ (A B) ≥ min {aii···i bii···i − Ri (A)Ri (B)} ,
(3.7)
1≤i≤n
where Ri (A) =
aii2 ···im , Ri (B) =
(i2 ,...,im )=(i,...,i)
bii2 ···im .
(i2 ,...,im )=(i,...,i)
Proof. Let λ = τ (A B). By Lemma 2.1, there exists i0 (1 ≤ i0 ≤ n) such that
|λ − ai0 i0 ···i0 bi0 i0 ···i0 | ≤
|ai0 i2 ···im bi0 i2 ···im |.
(i2 ,...,im )=(i0 ,...,i0 )
Then
λ ≥ ai0 i0 ···i0 bi0 i0 ···i0 −
ai0 i2 ···im bi0 i2 ···im
(i2 ,...,im )=(i0 ,...,i0 )
⎛
⎞⎛
≥ ai0 i0 ···i0 bi0 i0 ···i0 − ⎝
ai0 i2 ···im ⎠ ⎝
(i2 ,...,im )=(i0 ,...,i0 )
≥ min {aii···i bii···i − Ri (A)Ri (B)} . 1≤i≤n
⎞
bi0 i2 ···im ⎠
(i2 ,...,im )=(i0 ,...,i0 )
2
Corollary 3.8. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then A B is an M-tensor if aii···i bii···i ≥ Ri (A)Ri (B) for all i ∈ [n]. Proof. The conclusion follows from Theorem 3.7 and Lemma 2.5.
2
Next we give a numerical example and a table to compare the bounds of τ (A B) in Inequalities (3.1) and (3.3)–(3.7). Example 3.9. In this example, a tensor A = (ai1 i2 i3 i4 ) ∈ R[4,2] is written as an unfolded form a1111 a1211 a1112 a1212 a1121 a1221 a1122 a1222 . A= a2111 a2211 a2112 a2212 a2121 a2221 a2122 a2222 Some Z-tensors Ai , Bi ∈ R[4,2] (i = 1, 2, 3) are given as follows A1 = B1 =
3 0 −2 0
1.5 −0.5
0 0
0 0
0 0
0 0 0 0
0 0 0 0 0 0
0 0
0 0
−1 , 2
0 −1 , −1.5 2.5
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Table 1 Bounds for the minimum H-eigenvalue of the Fan product of Z-tensors. τ (A) τ (B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B)
of of of of of of
(3.1) (3.3) (3.4) (3.5) (3.6) (3.7)
A = A1 , B = B1
A = A2 , B = B2
A = A3 , B = B 3
1 0.5 2.7192
0.6525 0.4158 10.3834
0.6972 0.3234 9.8147
[2, +∞) [1, 5.5] [1.5, 6.5] [1.5, 5.5] [1.0359, 4.2929] [1, +∞)
[3.3967, +∞) [9.9326, 13.8163] [9.0125, 15.2100] [9.9326, 13.8163] [9.5122, 15.2100] [3.61, +∞)
[3.1324, +∞) [8.9711, 12.6647] [8.7675, 12.9897] [8.9711, 12.6647] [8.8713, 12.8560] [1.9, +∞)
0 −0.2 , −1 3.9 −1 −0.5 −0.9 −0.5 −0.1 0 3.2 −0.2 , B2 = 0 −0.2 −0.7 3.9 −0.6 −0.2 −0.7 −0.5 3.8 −0.6 −0.6 −0.4 −0.2 −0.5 −0.3 −0.6 , A3 = −0.9 −0.4 −0.2 −0.6 −0.4 −0.1 −0.3 3.7
A2 =
B3 =
−0.7 −0.6
3.8 −0.8 −0.4 −0.8
3.5 −0.3
−0.1 −0.6
−0.3 −0.4
−0.2 −0.4 −0.3 −0.5
−0.1 −0.4
−0.5 −0.4
−0.8 −0.1 −0.4 −0.3
−0.5 −0.5 . −0.9 3.1
For a Z-tensor A = sI − C (s ∈ R, C ≥ 0), we know that τ (A) = s − ρ(C). Let Ak = sk I − Ck (resp. Bk = tk I − Dk ). By the ZQW algorithm (see [27]), we get the value of ρ(Ck ) (resp. ρ(Dk )). And then we obtain τ (Ak ) (resp. τ (Bk )) (k = 1, 2, 3). According to Inequalities (3.1) and (3.3)–(3.7), we have Table 1. In the following, we give a lower bound on the minimum H-eigenvalue of the Fan product of s Z-tensors (s ≥ 2). When s = 2 and p1 = p2 = 1, the following inequality is Inequality (3.1). Theorem 3.10. Let Ak ∈ R[m,n] be Z-tensors, and let pk be positive integers with
s k=1
1 pk
1, where k = 1, 2, . . . , s. Then τ (A1 A2 · · · As ) ≥ min
1≤i≤n
(pk )
where Ak
s
(Ak )ii···i −
k=1
= A k A k · · · Ak . pk
Proof. We divide this proof into two cases. Case 1. A1 A2 · · · As is weakly irreducible.
s k=1
(p ) (Ak k )ii···i
−
p1
(p ) τ (Ak k )
k
,
≥
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
(pk )
As Ak is weakly irreducible, k ∈ [s]. Hence, Ak
67
= Ak Ak · · · Ak is weakly pk
irreducible. Similarly to the proof of Theorem 3.3(1), there exists xk ∈ Rn++ such that (pk )
Ak
[p ]
(pk )
(xk k )m−1 = τ (Ak
[p ]
)(xk k )[m−1] ,
[p ]
where xk k = ((xk )p1k , (xk )p2k , . . . , (xk )pnk )T ∈ Rn . Thus for any i ∈ [n], we have (p ) (p ) p (m−1) (Ak )piik2 ···im (xk )pi2k · · · (xk )pimk = τ (Ak k ) − (Ak k )ii···i (xk )i k .
(i2 ,...,im )=(i,...,i)
Denote C = A1 A2 · · · As and z = (z1 , z2 , . . . , zn )T > 0, where zi =
s
(xk )i . By
k=1
Lemma 2.7, we have
Cz m−1
i
zim−1 =
s
(Ak )ii···i −
k=1
≥
s
(Ak )ii···i −
k=1
=
=
s
(Ak )ii···i −
⎛ 1 ⎝ zim−1 (i 1
s
zim−1
k=1
1
k=1
zim−1
s
s
k=1
(Ak )ii···i −
s
⎞ (|(Ak )ii2 ···im | (xk )i2 · · · (xk )im )⎠
2 ,...,im )=(i,...,i) k=1
⎛
⎞ p1
⎝
k
pk ⎠
(|(Ak )ii2 ···im | (xk )i2 · · · (xk )im )
(i2 ,...,im )=(i,...,i)
s 1 (p ) (p ) p (m−1) pk (Ak k )ii···i − τ (Ak k ) (xk )i k k=1 (pk )
(Ak
(pk )
)ii···i − τ (Ak
p1
)
k
.
k=1
From Inequality (2.1), we have τ (A1 A2 · · · As ) ≥ min
1≤i≤n
s k=1
s p1 (p ) (p ) (Ak )ii···i − (Ak k )ii···i − τ (Ak k ) k
.
k=1
Case 2. A1 A2 · · · As is weakly reducible. The proof is similar to Case 2 of Theorem 3.3(1). 2 4. The positive definiteness for the Fan product of Z-tensors By using the inequalities in Section 3, we give some criteria for M-tensors and the positive definiteness of the Fan product of Z-tensors. And some properties for the determinant of the Fan product of M-tensors are obtained.
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Theorem 4.1. Let A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. (1) If τ (A)τ (B) ≤ (<) min {aii···i τ (B) + bii···i τ (A)}, then A B is an (resp. a nonsin1≤i≤n
gular) M-tensor. (2) Suppose A B is symmetric and m is even. If min {aii···i τ (B) + bii···i τ (A)} > τ (A)τ (B),
1≤i≤n
then A B is positive definite. (3) If A and B are nonsingular M-tensors, then A B − τ (A)τ (B)I is an M-tensor. (4) Suppose A B is symmetric and m is even. If max {Γ1 (A, B), Γ1 (B, A)} > 0, then A B is positive definite, where Γ1 (A, B) is defined before Theorem 3.6. Proof. (1) By Theorem 3.3(1), we have τ (A B) ≥ 0 (resp. > 0). The conclusion follows from Lemma 2.5. (2) By (1), we know that A B is a nonsingular M-tensor. Since A B is symmetric and m is even, we have A B is positive definite. (3) By Theorem 3.7 in [26], we have A B − τ (A B)I is an M-tensor. Then it follows from Theorem 3.3(2) that A B − τ (A B)I ≤ A B − τ (A)τ (B)I holds. And from Lemma 2.6, we obtain A B − τ (A)τ (B)I is an M-tensor. (4) By Theorem 3.6(3) and Lemma 2.5, we have A B is a nonsingular M-tensor. Similarly to the proof of (2), we get A B is positive definite. 2 Next, we use Theorem 4.1(2) to determine the positive definiteness of a homogeneous polynomial. Example 4.2. Let f (x) = x41 + x42 − 12 x31 x2 = Cx4 , where C = (cijkl ) ∈ R[4,2] with c1111 = c2222 = 1, c1112 = c1121 = c1211 = c2111 = − 18 , and the other entries are zeros. Let C be the Fan product of A = (aijkl ) ∈ R[4,2] and B = (bijkl ) ∈ R[4,2] , where a1111 = a2222 = b1111 = b2222 = 1, a1112 = a1121 = a1211 = a2111 = − 12 , b1112 = b1121 = b1211 = b2111 = − 14 , and the other entries are zeros. By Equation (1.1), we obtain {1, −0.14, 2.14} and {1, 0.43, 1.57} are the sets of the H-eigenvalues of A and B, respectively. Then τ (A) = −0.14 and τ (B) = 0.43. And by Theorem 4.1(2), we have f (x) is positive definite for all x = 0. Let A be a nonsingular M-tensor and B be an M-tensor. By Lemma 2.5 and Theorem 3.3(2), we have τ (B) ≤ τ (AB) τ (A) . When A is sparse and B is dense, A B is sparse. Thus we can get an upper bound on the minimum H-eigenvalue of a dense M-tensor by the minimum H-eigenvalue of a sparse M-tensor. We give the following example to show the upper bound of τ (B).
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
Example 4.3. Let A be the same as A1 in Example 3.9, and 1.5 −0.06 −0.03 −0.08 −0.07 −0.04 B= −0.4 −0.2 −0.2 −0.3 −0.5 −0.4
69
−0.12 −1 . −0.3 2.5
We know that τ (A) = 1 from Table 1. Then A is a nonsingular M-tensor. Since a diagonally dominant Z-tensor with nonnegative diagonal entries is an M-tensor [26], we have B is an M-tensor. Let C = A B. Then c1111 = 4.5, c2222 = 5, c1222 = −1, c2111 = −1, and the other entries are zeros. From Equation (1.1), we obtain {3.7192, 5.7808} is the set of the H-eigenvalues of C. Then τ (C) = 3.7192. By Theorem 3.3(2), we have (C) τ (B) ≤ ττ(A) = 3.7192. Let A be an m-order n-dimensional tensor with m ≥ 2. The determinant of A, denoted by det(A), is the resultant of the homogeneous polynomial system Axm−1 = 0. It is known that eigenvalues of A are roots of det(λI − A) (see [19]). It is complicated to compute the determinant of tensors. Here, we give some properties for the determinant of the Fan product of M-tensors. Theorem 4.4. Let A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] be M-tensors. (1) Then
d
|det (A B)| ≥ (τ (A B)) ≥
d min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B)
1≤i≤n
d
≥ (τ (A)τ (B)) , n−1
where d = n (m − 1) . (2) If det (A B) = 0. Then det (A) = 0 or det (B) = 0. (3) If det (A B) = 0 and aii···i bii···i = 0 for all i ∈ [n]. Then det (A) = 0 and det (B) = 0. (4) If A is a singular M-tensor and B is a nonsingular M-tensor, and aii···i > 0 for all i ∈ [n]. Then |det (A B)| > 0. n−1
Proof. (1) We know that the number of eigenvalues of A B is d = n (m − 1) and the product of all the eigenvalues is equal to det(A B) (see [19]). Since τ (A B) = d min {Re(λ) : λ ∈ σ(A B)}, we have |det (A B)| ≥ (τ (A B)) . By Theorem 3.3(2), the proof of (1) is completed. (2) Since det (A B) = 0, by (1) and Lemma 2.5(1), we have τ (A) = 0 or τ (B) = 0. Hence, det (A) = 0 or det (B) = 0. (3) Since det (A B) = 0, by (2), we have τ (A) = 0 or τ (B) = 0. Without loss of generality, suppose τ (A) = 0. By (1), we obtain min {aii···i τ (B)} = 0. Since aii···i = 0 1≤i≤n
for all i ∈ [n], we get τ (B) = 0. Hence, det (B) = 0. (4) By Lemma 2.5, we have τ (A) = 0 and τ (B) > 0. Since aii···i > 0 for all i ∈ [n], by Theorem 4.4(1), we have |det (A B)| > 0. 2
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Declaration of Competing Interest The authors have no competing interests for the research conducted in this paper. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 11371109 and No. 11801115), the Natural Science Foundation of Heilongjiang Province (No. QC2018002). References [1] B.D. Anderson, E.I. Jury, Stability of multidimensional digital filters, IEEE Trans. Circuits Syst. 21 (1974) 300–304. [2] B.D. Anderson, N.K. Bose, E.I. Jury, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control 20 (1975) 55–66. [3] N.K. Bose, P.S. Kamat, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process. 22 (1974) 307–314. [4] N.K. Bose, R.W. Newcomb, Tellegon’s theorem and multivariable realizability theory, Int. J. Electron. 36 (1974) 417–425. [5] G. Cheng, New bounds for the minimum eigenvalue of the Fan product of two M-matrices, Czechoslovak Math. J. 64 (2014) 63–68. [6] W. Ding, L. Qi, Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl. 439 (2013) 3264–3278. [7] K. Fan, Inequalities for M-matrices, Indag. Math. 67 (1964) 602–610. [8] M. Fang, Bounds on eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 425 (2007) 7–15. [9] S. Friedland, S. Gaubert, L. Han, Perron-Frobenius theorems for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013) 738–749. [10] M.S. Gowda, Z. Luo, L. Qi, N. Xiu, Z-tensors and complementarity problems, arXiv:1510.07933v2. [11] G.H. Hardy, J.E. Littlewood, G. Polya, Inequality, Cambridge University Press, Cambridge, 1934. [12] J. He, T. Huang, Inequalities for M-tensors, J. Inequal. Appl. 2014 (2014) 114. [13] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. [14] S. Hu, Z. Huang, L. Qi, Strictly nonnegative tensors and nonnegative tensor partition, Sci. China Math. 57 (2014) 181–195. [15] M. Kannan, N. Shaked-Monderer, A. Berman, Some properties of strong H-tensors and general H-tensors, Linear Algebra Appl. 476 (2015) 42–55. [16] M. Kannan, N. Shaked-Monderer, A. Berman, On weakly irreducible nonnegative tensors and interval hull of some classes of tensors, Linear Multilinear Algebra 64 (2016) 667–679. [17] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, in: Proceedings 1st IEEE International Workshop on Computational Advances of Multisensor Adaptive Processing, 2005, pp. 129–132. [18] Q. Ni, L. Qi, F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control 53 (2008) 1096–1107. [19] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302–1324. [20] L. Qi, Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. [21] J. Shao, L. You, On some properties of three different types of triangular blocked tensors, Linear Algebra Appl. 511 (2016) 110–140. [22] L. Sun, B. Zheng, J. Zhou, H. Yan, Some inequalities for the Hadamard product of tensors, Linear Multilinear Algebra 66 (2018) 1199–1214. [23] G. Wang, Y. Wang, Y. Zhang, Some inequalities for the Fan product of M-tensors, J. Inequal. Appl. 2018 (2018) 257. [24] Y. Yang, Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl. 31 (2010) 2517–2530.
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[25] Y. Yang, Q. Yang, On some properties of nonnegative weakly irreducible tensors, arXiv:1111.0713v2. [26] L. Zhang, L. Qi, G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl. 35 (2014) 437–452. [27] G. Zhou, L. Qi, S.-Y. Wu, Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor, Front. Math. China 8 (2013) 155–168. [28] J. Zhou, L. Sun, Y. Wei, C. Bu, Some characterizations of M-tensors via digraphs, Linear Algebra Appl. 495 (2016) 190–198.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Some inequalities on the minimum H-eigenvalue of the Fan product of Z-tensors Jihong Shen a,b , Yue Wang a , Changjiang Bu a,b,∗ a
College of Automation, Harbin Engineering University, Harbin 150001, PR China College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, PR China b
a r t i c l e
i n f o
Article history: Received 23 November 2018 Accepted 24 May 2019 Available online 29 May 2019 Submitted by J. Shao MSC: 15A69 15A18 65F15
a b s t r a c t In this paper, some bounds on the minimum H-eigenvalue of a Z-tensor and the Fan product of Z-tensors are given. By applying the bounds, we obtain some criteria for M-tensors and the positive definiteness of the Fan product of Z-tensors, and give some properties for the determinant of the Fan product of M-tensors. © 2019 Elsevier Inc. All rights reserved.
Keywords: Z-tensor H-eigenvalue Fan product Positive definiteness
1. Introduction For a positive integer n, let [n] = {1, 2, . . . , n}. An m-order n-dimensional tensor A = (ai1 i2 ···im ) (ij ∈ [n] , j ∈ [m]) is a multidimensional array with nm entries. We sometimes write ai1 i2 ···im as Ai1 i2 ···im . When m = 2, A is an n × n matrix. The tensor A is called nonnegative if all its entries are nonnegative, denoted by A ≥ 0. A tensor whose diagonal * Corresponding author. E-mail address: [email protected] (C. Bu). https://doi.org/10.1016/j.laa.2019.05.031 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
entries are ones and off-diagonal entries are zeros is called the unit tensor, denoted by I. Let C [m,n] and R[m,n] be the sets of m-order n-dimensional complex tensors and real tensors, respectively. Denote C n (resp. Rn ) the set of n-dimensional vectors over C (resp. R). In 2005, the concept of eigenvalues of tensors was proposed by Qi [19] and Lim [17], independently. For a tensor A = (ai1 i2 ···im ) ∈ C [m,n] , λ ∈ C is called an eigenvalue of A T if there exists a nonzero vector x = (x1 , . . . , xn ) ∈ C n such that Axm−1 = λx[m−1] ,
(1.1)
where Axm−1 is a vector in C n whose i-th component is (Axm−1 )i =
n
aii2 ···im xi2 · · · xim (i = 1, . . . , n),
i2 ,...,im =1
T and x[m−1] = xm−1 , ..., xm−1 . If λ ∈ R and x ∈ Rn , then λ is called an Hn 1 eigenvalue of A. Let σ(A) denote the set of all the eigenvalues of A. Denote ρ(A) = max {|λ| : λ ∈ σ(A)} the spectral radius of A and τ (A) = min {Re(λ) : λ ∈ σ(A)}. A tensor A = (ai1 i2 ···im ) is called symmetric if entries ai1 i2 ···im are invariant under any permutation of indices i1 · · · im . An m-th degree homogeneous polynomial f (x) can be represented by a symmetric tensor A as f (x) = Axm :=
n
ai1 i2 ···im xi1 xi2 · · · xim .
i1 ,i2 ,...,im =1
A symmetric tensor A = (ai1 i2 ···im ) ∈ R[m,n] is called positive definite (resp. positive semidefinite) if f (x) > 0 (resp. ≥ 0) for all 0 = x ∈ Rn . When A is a positive definite tensor, m must be even [19]. The positive definiteness of a homogeneous polynomial plays an important role in the stability study of nonlinear systems via Lyapunov’s direct method [18], such as tests for filters [1,3], the multivariate network realizability theory [4], and the output feedback stabilization problems [2]. In [19], Qi proved that a real symmetric tensor A is positive definite if and only if the minimum H-eigenvalue of A is positive. A tensor A ∈ R[m,n] is called a Z-tensor if all of its off-diagonal entries are nonpositive. A Z-tensor A = sI −B is called an M-tensor (resp. a nonsingular M-tensor) if s ≥ ρ(B) (resp. s > ρ(B)), where s ∈ R and B ≥ 0. Clearly, M-tensors are Z-tensors. Zhang et al. proved that a symmetric Z-tensor is a nonsingular M-tensor if and only if it is positive definite [26]. Gowda et al. showed that τ (A) is the minimum H-eigenvalue of a Z-tensor A (see [10]). A Z-tensor A is a nonsingular M-tensor if and only if τ (A) > 0 (see [26]). Some bounds on the minimum H-eigenvalue of Z-tensors and M-tensors were given [12, 22]. More results on M-tensors can be found in [6,20,21,28].
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For A = (aij ), B = (bij ) ∈ R[2,n] , the Fan product A B is a matrix with entries (A B)ij = (−1)δij +1 aij bij , where δij = 1 if i = j, and δij = 0 otherwise. In 1964, this product was proposed by Ky Fan and it was proved that this product of M -matrices is also an M -matrix [7]. Some bounds on τ (A B) for M -matrices A and B were given by τ (A), τ (B) or the entries of A and B [5,8,13]. It is natural to study the Fan product of tensors. For A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] , the Fan product A B is a tensor with entries (A B)i1 i2 ···im = (−1)δi1 i2 ···im +1 ai1 i2 ···im bi1 i2 ···im , where δi1 i2 ···im = 1 if i1 = i2 = · · · = im , and δi1 i2 ···im = 0 otherwise [15]. Clearly, the Fan product of Z-tensors is a Z-tensor. In this paper, we give some bounds on the minimum H-eigenvalue of a Z-tensor. And some bounds on the minimum H-eigenvalue τ (A B) for Z-tensors A and B are established with τ (A) and τ (B). Further, we use the bounds to give some criteria to determine M-tensors and the positive definiteness of the Fan product of Z-tensors. And some properties for the determinant of the Fan product of M-tensors are obtained. 2. Preliminaries In this section, we introduce some lemmas which are used to obtain the main results in Sections 3 and 4. Lemma 2.1. [19] Let A = (ai1 i2 ···im ) ∈ C [m,n] . Then σ(A) ⊆
n
{z ∈ C : |z − ai···i | ≤ Ri (A)},
i=1
where Ri (A) =
|aii2 ···im |.
(i2 ,...,im )=(i,...,i)
Denote Rn++ the set of n-dimensional positive vectors. The following lemma is Lemma 4.1 in [22]. Lemma 2.2. [22] Let A = (ai1 ···im ) ∈ R[m,n] be a Z-tensor. Then (Axm−1 )i ≤ τ (A) ≤ min aii···i 1≤i≤n 1≤i≤n xm−1 i min
T
for any x = (x1 , . . . , xn ) ∈ Rn++ . Using similar proof of the above lemma given in [22], we have Inequality (2.2).
(2.1)
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Lemma 2.3. Let A = (ai1 ···im ) ∈ R[m,n] be a Z-tensor. Then (Axm−1 )i 1≤i≤n xm−1 i
τ (A) ≤ max
(2.2)
T
for any x = (x1 , . . . , xn ) ∈ Rn++ . T
Proof. Let A = sI − B, where s ∈ R and B ≥ 0. For any x = (x1 , . . . , xn ) ∈ Rn++ , we obtain sxm−1 (Bxm−1 )i (Axm−1 )i − (Bxm−1 )i i = = s − . xm−1 xm−1 xm−1 i i i By the definition of eigenvalues of tensors, we know that τ (A) = s − ρ(B) (see [10]). Lemma 5.3 in [24] shows that min 1≤i≤n (Axm−1 )i s − min s − xm−1 . Hence 1≤i≤n
Bxm−1 xim−1
i
≤ ρ(B) ≤ max
1≤i≤n
Bxm−1 xim−1
i
. Then s − ρ(B) ≤
i
(Axm−1 )i . 1≤i≤n xm−1 i
τ (A) ≤ max
2
For A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] , A ≤ B means that ai1 ···im ≤ bi1 ···im for all ij ∈ [n] , j ∈ [m]. Lemma 2.4. Let {Ak } be a sequence of Z-tensors satisfying Ak ≤ Ak+1 for each positive integer k, and lim Ak = A. Then k→∞
lim τ (Ak ) = τ (A).
k→∞
Proof. We know that there exist s ∈ R, Bk ≥ 0 and B ≥ 0 such that Ak = sI − Bk and A = sI − B. So τ (Ak ) = s − ρ(Bk ). Since Ak ≤ Ak+1 and lim Ak = A, we have k→∞
Bk+1 ≤ Bk and lim Bk = B. Proposition 3.1 in [22] gives that lim ρ(Bk ) = ρ(B). Then k→∞
k→∞
lim τ (Ak ) = lim (s − ρ(Bk )) = s − lim ρ(Bk ) = s − ρ(B) = τ (A).
k→∞
k→∞
k→∞
2
For a Z-tensor A, Lemma 2.5 is a criterion to determine that A is an (resp. a nonsingular) M-tensor. Lemma 2.5. [26] Let A ∈ R[m,n] be a Z-tensor. Then (1) A is an M-tensor if and only if τ (A) ≥ 0; (2) A is a nonsingular M-tensor if and only if τ (A) > 0.
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
59
Lemma 2.6. [12] Let A and B be Z-tensors satisfying A ≤ B. If A is an M-tensor, then B is an M-tensor and τ (A) ≤ τ (B). The following inequality is a transformation of Hölder’s Inequality, which is used to prove Theorem 3.10. Lemma 2.7. [11] If pj > 0 (j ∈ [s]) and
s j=1
where xji ≥ 0, i ∈ [n].
1 pj
≥ 1, then
s n
x ji ≤
i=1 j=1
s j=1
n i=1
p xjij
p1
j
,
3. Bounds for the minimum H-eigenvalue of Z-tensors In this section, we give some bounds on the minimum H-eigenvalue of a Z-tensor and the Fan product of Z-tensors. The lower bounds can be used to determine Z-tensors are (nonsingular) M-tensors and the positive definiteness of Z-tensors. For a set α, |α| denotes the cardinality of α. Let A = (ai1 ···im ) ∈ C [m,n] and α ⊂ [n] with |α| = r. A principal subtensor of A with index set α is A[α] = (ai1 ···im ), where i1 , . . . , im ∈ α (see [16]). In [14], Hu et al. proved that ρ(A[α]) ≤ ρ(A). For a Z-tensor A = sI − B, we know that τ (A) = s − ρ(B), where s ∈ R and B ≥ 0. Hence, τ (A[α]) = s − ρ(B[α]), where B[α] is a principal subtensor of B. Since ρ(B[α]) ≤ ρ(B), we obtain the following theorem. Theorem 3.1. Let A ∈ R[m,n] be a Z-tensor. Then τ (A) ≤ τ (A[α]), where α ⊂ [n]. Let Rmin (A) = min
n
1≤i≤n i2 ,...,im =1
aii2 ···im and Rmax (A) = max
n
1≤i≤n i2 ,...,im =1
aii2 ···im be the
smallest and the largest row sums of a tensor A = (ai1 i2 ···im ) ∈ R[m,n] , respectively. A tensor A is called weakly reducible, if there exists some nonempty proper subset I ⊂ [n] such that ai1 i2 ···im = 0 for any i1 ∈ I and at least an ij ∈ [n]\I, j = 2, . . . , m. Otherwise, A is called weakly irreducible [9,25]. If A is nonnegative weakly irreducible, then ρ(A) is an eigenvalue of A and there exists a positive eigenvector corresponding to ρ(A) (see [25]). For an irreducible M-tensor A, we know that Rmin (A) ≤ τ (A) ≤ Rmax (A) (see [12]). We extend this result to general Z-tensors. Theorem 3.2. Let A = (ai1 i2 ···im ) ∈ R[m,n] be a Z-tensor. Then Rmin (A) ≤ τ (A) ≤ Rmax (A) .
60
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Proof. A Z-tensor A can be written as A = sI − B, where s ∈ R and B ≥ 0. Then Rmin (A) = s −Rmax (B) and Rmax (A) = s −Rmin (B). Since Rmin (B) ≤ ρ (B) ≤ Rmax (B) (see Lemma 5.2 of [24]), we have Rmin (A) ≤ τ (A) ≤ Rmax (A). 2 For M -matrices A = (aij ), B = (bij ) ∈ R[2,n] , Horn and Johnson obtained that τ (A B) ≥ τ (A)τ (B) (see [13]). Fang proved that τ (A B) ≥ min {aii τ (B) + bii τ (A)}− 1≤i≤n
τ (A)τ (B) ≥ τ (A)τ (B) (see [8]). We extend these results to tensors. Theorem 3.3. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be two tensors. (1) If A and B are Z-tensors, then τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B).
(3.1)
1≤i≤n
(2) If A and B are M-tensors, then τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B) ≥ τ (A)τ (B). 1≤i≤n
Proof. (1) We divide this proof into two cases. Case 1. A B is weakly irreducible. By the definition of the Fan product of tensors, we know that A and B are weakly irreducible. A Z-tensor A can be written as A = tI −D and τ (A) = t −ρ(D), where t ∈ R and D ≥ 0. Since D is nonnegative weakly irreducible, there exists a positive eigenvector corresponding to ρ(D). And this vector is also a positive eigenvector corresponding to τ (A). Then there exists x = (x1 , x2 , . . . , xn )T ∈ Rn++ such that Axm−1 = τ (A)x[m−1] . Similarly, there exists y = (y1 , y2 , . . . , yn )T ∈ Rn++ such that By m−1 = τ (B)y [m−1] . Hence (τ (A) − aii···i ) xm−1 = i
aii2 ···im xi2 · · · xim (i = 1, 2, . . . , n)
(i2 ,...,im )=(i,...,i)
and (τ (B) − bii···i ) yim−1 =
bii2 ···im yi2 · · · yim (i = 1, 2, . . . , n) .
(i2 ,...,im )=(i,...,i)
Let z = (z1 , z2 , . . . , zn )T ∈ Rn++ , where zi = xi yi for all i ∈ [n]. Then (A B) z m−1 i zim−1 = aii···i bii···i −
⎛ 1 zim−1
⎝
(i2 ,...,im )=(i,...,i)
⎞ |aii2 ···im | |bii2 ···im | xi2 yi2 · · · xim yim ⎠
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
⎛ ≥ aii···i bii···i − ⎛ ×⎝
1 zim−1
⎝
61
⎞
|aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎞
|bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i)
= aii···i bii···i − (aii···i − τ (A)) (bii···i − τ (B)) = aii···i τ (B) + bii···i τ (A) − τ (A)τ (B). By Lemma 2.2, we have τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B).
(3.2)
1≤i≤n
Case 2. A B is weakly reducible. Let Ak = A + k1 T and Bk = B + k1 T , where k is a positive integer and T = (ti1 i2 ···im ) is a tensor whose diagonal entries are zeros and off-diagonal entries are negative. Then Ak and Bk are weakly irreducible Z-tensors. And Ak and Bk are monotone increasing sequences with lim Ak = A and lim Bk = B, respectively. By Lemma 2.4, we obtain k→∞
k→∞
lim τ (Ak ) = τ (A) and lim τ (Bk ) = τ (B). Substituting Ak and Bk into Inequality
k→∞
k→∞
(3.2), and let k → ∞, Inequality (3.1) holds. (2) By Statement (1), we get τ (A B) ≥ min {aii···i τ (B) + bii···i τ (A)}−τ (A)τ (B). Since 1≤i≤n
A and B are M-tensors, by Lemma 2.2 and Lemma 2.5(1), we have τ (A) ≤ min aii···i , 1≤i≤n
τ (B) ≤ min bii···i , τ (A) ≥ 0, and τ (B) ≥ 0. Then we obtain the inequality of (2). 2 1≤i≤n
Next we show that M-tensors (resp. nonsingular M-tensors) are closed under Fan product, which can be seen in [23]. Theorem 3.4. Let A = (ai1 ···im ), B = (bi1 ···im ) ∈ R[m,n] be M-tensors (resp. nonsingular M-tensors). Then A B is an M-tensor (resp. a nonsingular M-tensor). Proof. By Theorem 3.3(2) and Lemma 2.5, we have τ (A B) ≥ τ (A)τ (B) ≥ 0 (resp. > 0). Then A B is an M-tensor (resp. a nonsingular M-tensor). 2 By Theorem 3.1, Theorem 3.3(2) and Lemma 2.5, we can get the following result immediately. Theorem 3.5. (Theorem 4.1 in [16]) Let A = (ai1 ···im ), B = (bi1 ···im ) ∈ R[m,n] be M-tensors (resp. nonsingular M-tensors). Then all principal subtensors of A B are M-tensors (resp. nonsingular M-tensors).
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62
For A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] , let αi (A) = βi (A) =
min
(i2 ,...,im )=(i,...,i)
|aii2 ···im |, and
max
(i2 ,...,im )=(i,...,i)
|aii2 ···im |,
Γ1 (A, B) = min {(aii···i − αi (A)) bii···i + αi (A)τ (B)} , 1≤i≤n
Γ2 (A, B) = max {(aii···i − βi (A)) bii···i + βi (A)τ (B)} , 1≤i≤n
1 1 1 1 Γ3 (A, B) = min aii···i bii···i − (αi (A)) 2 (αi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 , 1≤i≤n
Γ4 (A, B) = max
1≤i≤n
1 1 1 1 aii···i bii···i − (βi (A)) 2 (βi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 .
Here, we give new lower and upper bounds of τ (A B). Theorem 3.6. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then (1) Γ1 (A, B) ≤ τ (A B) ≤ Γ2 (A, B),
(3.3)
Γ1 (B, A) ≤ τ (A B) ≤ Γ2 (B, A),
(3.4)
max {Γ1 (A, B), Γ1 (B, A)} ≤ τ (A B) ≤ min {Γ2 (A, B), Γ2 (B, A)} ,
(3.5)
Γ3 (A, B) ≤ τ (A B) ≤ Γ4 (A, B).
(3.6)
(2)
(3)
(4)
Proof. (1) We prove the result under two cases. Case 1. B is weakly irreducible. Similarly to the proof of Theorem 3.3(1), there exists u = (u1 , u2 , . . . , un )T ∈ Rn++ such that Bum−1 = τ (B)u[m−1] . Then (τ (B) − bii···i ) um−1 = i
bii2 ···im ui2 · · · uim , i ∈ [n] .
(i2 ,...,im )=(i,...,i)
Hence (A B) um−1 i um−1 i
− aii···i bii···i um−1 i =
|aii2 ···im | |bii2 ···im | ui2 · · · uim
(i2 ,...,im )=(i,...,i) um−1 i
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
⎛ αi (A) ⎝ ≥ aii···i bii···i − m−1 ui (i
63
⎞
|bii2 ···im | ui2 · · · uim ⎠
2 ,...,im )=(i,...,i)
= aii···i bii···i − αi (A) (bii···i − τ (B)) = (aii···i − αi (A)) bii···i + αi (A)τ (B) and
(A B) um−1
i
um−1 i
⎛ βi (A) ⎝ ≤ aii···i bii···i − m−1 ui (i
⎞ |bii2 ···im | ui2 · · · uim ⎠
2 ,...,im )=(i,...,i)
= aii···i bii···i − βi (A) (bii···i − τ (B)) = (aii···i − βi (A)) bii···i + βi (A)τ (B). By Inequalities (2.1) and (2.2), we have Γ1 (A, B) ≤ τ (A B) ≤ Γ2 (A, B). Case 2. B is weakly reducible. By the proof of Case 2 in Theorem 3.3(1), we have Inequality (3.3). As the Fan product has the commutative law, Inequality (3.4) holds. From Inequalities (3.3) and (3.4), we obtain Inequality (3.5). (4) We divide this proof into two cases. Case 1. A B is weakly irreducible. Since A and B are weakly irreducible, similarly to the proof of Theorem 3.3(1), there exist x = (x1 , x2 , . . . , xn )T ∈ Rn++ and y = (y1 , y2 , . . . , yn )T ∈ Rn++ such that Axm−1 = τ (A)x[m−1] and By m−1 = τ (B)y [m−1] . Then
(τ (A) − aii···i ) xm−1 = i
aii2 ···im xi2 · · · xim ,
(i2 ,...,im )=(i,...,i)
(τ (B) − bii···i ) yim−1 =
bii2 ···im yi2 · · · yim .
(i2 ,...,im )=(i,...,i)
By Cauchy-Schwarz Inequality, we have
1
1
1
1
|aii2 ···im | |bii2 ···im | xi22 yi22 · · · xi2m yi2m
(i2 ,...,im )=(i,...,i)
⎛ ≤ ⎝
(i2 ,...,im )=(i,...,i)
⎞ 12 ⎛ a2ii2 ···im xi2 · · · xim ⎠ ⎝
(i2 ,...,im )=(i,...,i)
⎞ 12 b2ii2 ···im yi2 · · · yim ⎠
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
64
⎛
⎞ 12
≤⎝
αi (A) |aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
×⎝
αi (B) |bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
= (αi (A)) (αi (B)) ⎝ 1 2
1 2
|aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎛
⎞ 12
×⎝
|bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i) 1
m−1 2
1
1
= (αi (A)) 2 (αi (B)) 2 (aii···i − τ (A)) 2 xi
1
m−1 2
(bii···i − τ (B)) 2 yi
1
.
1
Let z = (z1 , z2 , . . . , zn )T ∈ Rn++ , where zi = xi2 yi2 for all i ∈ [n]. Hence (A B) z m−1 i zim−1
⎛
= aii···i bii···i −
1 zim−1
⎞
⎝
1 2 i2
1 2 i2
1 2 im
1 2 im
|aii2 ···im | |bii2 ···im | x y · · · x y ⎠
(i2 ,...,im )=(i,...,i) 1 2
1
1
1
≥ aii···i bii···i − (αi (A)) (αi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 and (A B) z m−1 i zim−1 ≤ aii···i bii···i − ⎛ ×⎝
⎛ 1 zim−1
(βi (A)) (βi (B)) ⎝
1 2
1 2
⎞ 12 |aii2 ···im | xi2 · · · xim ⎠
(i2 ,...,im )=(i,...,i)
⎞ 12 |bii2 ···im | yi2 · · · yim ⎠
(i2 ,...,im )=(i,...,i) 1
1
1
1
= aii···i bii···i − (βi (A)) 2 (βi (B)) 2 (aii···i − τ (A)) 2 (bii···i − τ (B)) 2 . From Inequalities (2.1) and (2.2), we obtain Inequality (3.6). Case 2. A B is weakly reducible. By the proof of Case 2 in Theorem 3.3(1), we get Inequality (3.6). 2
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
65
Next, we give a lower bound of τ (A B) with the entries of tensors A and B. Theorem 3.7. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then τ (A B) ≥ min {aii···i bii···i − Ri (A)Ri (B)} ,
(3.7)
1≤i≤n
where Ri (A) =
aii2 ···im , Ri (B) =
(i2 ,...,im )=(i,...,i)
bii2 ···im .
(i2 ,...,im )=(i,...,i)
Proof. Let λ = τ (A B). By Lemma 2.1, there exists i0 (1 ≤ i0 ≤ n) such that
|λ − ai0 i0 ···i0 bi0 i0 ···i0 | ≤
|ai0 i2 ···im bi0 i2 ···im |.
(i2 ,...,im )=(i0 ,...,i0 )
Then
λ ≥ ai0 i0 ···i0 bi0 i0 ···i0 −
ai0 i2 ···im bi0 i2 ···im
(i2 ,...,im )=(i0 ,...,i0 )
⎛
⎞⎛
≥ ai0 i0 ···i0 bi0 i0 ···i0 − ⎝
ai0 i2 ···im ⎠ ⎝
(i2 ,...,im )=(i0 ,...,i0 )
≥ min {aii···i bii···i − Ri (A)Ri (B)} . 1≤i≤n
⎞
bi0 i2 ···im ⎠
(i2 ,...,im )=(i0 ,...,i0 )
2
Corollary 3.8. Let A = (ai1 i2 ···im ), B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. Then A B is an M-tensor if aii···i bii···i ≥ Ri (A)Ri (B) for all i ∈ [n]. Proof. The conclusion follows from Theorem 3.7 and Lemma 2.5.
2
Next we give a numerical example and a table to compare the bounds of τ (A B) in Inequalities (3.1) and (3.3)–(3.7). Example 3.9. In this example, a tensor A = (ai1 i2 i3 i4 ) ∈ R[4,2] is written as an unfolded form a1111 a1211 a1112 a1212 a1121 a1221 a1122 a1222 . A= a2111 a2211 a2112 a2212 a2121 a2221 a2122 a2222 Some Z-tensors Ai , Bi ∈ R[4,2] (i = 1, 2, 3) are given as follows A1 = B1 =
3 0 −2 0
1.5 −0.5
0 0
0 0
0 0
0 0 0 0
0 0 0 0 0 0
0 0
0 0
−1 , 2
0 −1 , −1.5 2.5
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
66
Table 1 Bounds for the minimum H-eigenvalue of the Fan product of Z-tensors. τ (A) τ (B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B) τ (A B)
of of of of of of
(3.1) (3.3) (3.4) (3.5) (3.6) (3.7)
A = A1 , B = B1
A = A2 , B = B2
A = A3 , B = B 3
1 0.5 2.7192
0.6525 0.4158 10.3834
0.6972 0.3234 9.8147
[2, +∞) [1, 5.5] [1.5, 6.5] [1.5, 5.5] [1.0359, 4.2929] [1, +∞)
[3.3967, +∞) [9.9326, 13.8163] [9.0125, 15.2100] [9.9326, 13.8163] [9.5122, 15.2100] [3.61, +∞)
[3.1324, +∞) [8.9711, 12.6647] [8.7675, 12.9897] [8.9711, 12.6647] [8.8713, 12.8560] [1.9, +∞)
0 −0.2 , −1 3.9 −1 −0.5 −0.9 −0.5 −0.1 0 3.2 −0.2 , B2 = 0 −0.2 −0.7 3.9 −0.6 −0.2 −0.7 −0.5 3.8 −0.6 −0.6 −0.4 −0.2 −0.5 −0.3 −0.6 , A3 = −0.9 −0.4 −0.2 −0.6 −0.4 −0.1 −0.3 3.7
A2 =
B3 =
−0.7 −0.6
3.8 −0.8 −0.4 −0.8
3.5 −0.3
−0.1 −0.6
−0.3 −0.4
−0.2 −0.4 −0.3 −0.5
−0.1 −0.4
−0.5 −0.4
−0.8 −0.1 −0.4 −0.3
−0.5 −0.5 . −0.9 3.1
For a Z-tensor A = sI − C (s ∈ R, C ≥ 0), we know that τ (A) = s − ρ(C). Let Ak = sk I − Ck (resp. Bk = tk I − Dk ). By the ZQW algorithm (see [27]), we get the value of ρ(Ck ) (resp. ρ(Dk )). And then we obtain τ (Ak ) (resp. τ (Bk )) (k = 1, 2, 3). According to Inequalities (3.1) and (3.3)–(3.7), we have Table 1. In the following, we give a lower bound on the minimum H-eigenvalue of the Fan product of s Z-tensors (s ≥ 2). When s = 2 and p1 = p2 = 1, the following inequality is Inequality (3.1). Theorem 3.10. Let Ak ∈ R[m,n] be Z-tensors, and let pk be positive integers with
s k=1
1 pk
1, where k = 1, 2, . . . , s. Then τ (A1 A2 · · · As ) ≥ min
1≤i≤n
(pk )
where Ak
s
(Ak )ii···i −
k=1
= A k A k · · · Ak . pk
Proof. We divide this proof into two cases. Case 1. A1 A2 · · · As is weakly irreducible.
s k=1
(p ) (Ak k )ii···i
−
p1
(p ) τ (Ak k )
k
,
≥
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
(pk )
As Ak is weakly irreducible, k ∈ [s]. Hence, Ak
67
= Ak Ak · · · Ak is weakly pk
irreducible. Similarly to the proof of Theorem 3.3(1), there exists xk ∈ Rn++ such that (pk )
Ak
[p ]
(pk )
(xk k )m−1 = τ (Ak
[p ]
)(xk k )[m−1] ,
[p ]
where xk k = ((xk )p1k , (xk )p2k , . . . , (xk )pnk )T ∈ Rn . Thus for any i ∈ [n], we have (p ) (p ) p (m−1) (Ak )piik2 ···im (xk )pi2k · · · (xk )pimk = τ (Ak k ) − (Ak k )ii···i (xk )i k .
(i2 ,...,im )=(i,...,i)
Denote C = A1 A2 · · · As and z = (z1 , z2 , . . . , zn )T > 0, where zi =
s
(xk )i . By
k=1
Lemma 2.7, we have
Cz m−1
i
zim−1 =
s
(Ak )ii···i −
k=1
≥
s
(Ak )ii···i −
k=1
=
=
s
(Ak )ii···i −
⎛ 1 ⎝ zim−1 (i 1
s
zim−1
k=1
1
k=1
zim−1
s
s
k=1
(Ak )ii···i −
s
⎞ (|(Ak )ii2 ···im | (xk )i2 · · · (xk )im )⎠
2 ,...,im )=(i,...,i) k=1
⎛
⎞ p1
⎝
k
pk ⎠
(|(Ak )ii2 ···im | (xk )i2 · · · (xk )im )
(i2 ,...,im )=(i,...,i)
s 1 (p ) (p ) p (m−1) pk (Ak k )ii···i − τ (Ak k ) (xk )i k k=1 (pk )
(Ak
(pk )
)ii···i − τ (Ak
p1
)
k
.
k=1
From Inequality (2.1), we have τ (A1 A2 · · · As ) ≥ min
1≤i≤n
s k=1
s p1 (p ) (p ) (Ak )ii···i − (Ak k )ii···i − τ (Ak k ) k
.
k=1
Case 2. A1 A2 · · · As is weakly reducible. The proof is similar to Case 2 of Theorem 3.3(1). 2 4. The positive definiteness for the Fan product of Z-tensors By using the inequalities in Section 3, we give some criteria for M-tensors and the positive definiteness of the Fan product of Z-tensors. And some properties for the determinant of the Fan product of M-tensors are obtained.
68
J. Shen et al. / Linear Algebra and its Applications 579 (2019) 55–71
Theorem 4.1. Let A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] be Z-tensors. (1) If τ (A)τ (B) ≤ (<) min {aii···i τ (B) + bii···i τ (A)}, then A B is an (resp. a nonsin1≤i≤n
gular) M-tensor. (2) Suppose A B is symmetric and m is even. If min {aii···i τ (B) + bii···i τ (A)} > τ (A)τ (B),
1≤i≤n
then A B is positive definite. (3) If A and B are nonsingular M-tensors, then A B − τ (A)τ (B)I is an M-tensor. (4) Suppose A B is symmetric and m is even. If max {Γ1 (A, B), Γ1 (B, A)} > 0, then A B is positive definite, where Γ1 (A, B) is defined before Theorem 3.6. Proof. (1) By Theorem 3.3(1), we have τ (A B) ≥ 0 (resp. > 0). The conclusion follows from Lemma 2.5. (2) By (1), we know that A B is a nonsingular M-tensor. Since A B is symmetric and m is even, we have A B is positive definite. (3) By Theorem 3.7 in [26], we have A B − τ (A B)I is an M-tensor. Then it follows from Theorem 3.3(2) that A B − τ (A B)I ≤ A B − τ (A)τ (B)I holds. And from Lemma 2.6, we obtain A B − τ (A)τ (B)I is an M-tensor. (4) By Theorem 3.6(3) and Lemma 2.5, we have A B is a nonsingular M-tensor. Similarly to the proof of (2), we get A B is positive definite. 2 Next, we use Theorem 4.1(2) to determine the positive definiteness of a homogeneous polynomial. Example 4.2. Let f (x) = x41 + x42 − 12 x31 x2 = Cx4 , where C = (cijkl ) ∈ R[4,2] with c1111 = c2222 = 1, c1112 = c1121 = c1211 = c2111 = − 18 , and the other entries are zeros. Let C be the Fan product of A = (aijkl ) ∈ R[4,2] and B = (bijkl ) ∈ R[4,2] , where a1111 = a2222 = b1111 = b2222 = 1, a1112 = a1121 = a1211 = a2111 = − 12 , b1112 = b1121 = b1211 = b2111 = − 14 , and the other entries are zeros. By Equation (1.1), we obtain {1, −0.14, 2.14} and {1, 0.43, 1.57} are the sets of the H-eigenvalues of A and B, respectively. Then τ (A) = −0.14 and τ (B) = 0.43. And by Theorem 4.1(2), we have f (x) is positive definite for all x = 0. Let A be a nonsingular M-tensor and B be an M-tensor. By Lemma 2.5 and Theorem 3.3(2), we have τ (B) ≤ τ (AB) τ (A) . When A is sparse and B is dense, A B is sparse. Thus we can get an upper bound on the minimum H-eigenvalue of a dense M-tensor by the minimum H-eigenvalue of a sparse M-tensor. We give the following example to show the upper bound of τ (B).
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Example 4.3. Let A be the same as A1 in Example 3.9, and 1.5 −0.06 −0.03 −0.08 −0.07 −0.04 B= −0.4 −0.2 −0.2 −0.3 −0.5 −0.4
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−0.12 −1 . −0.3 2.5
We know that τ (A) = 1 from Table 1. Then A is a nonsingular M-tensor. Since a diagonally dominant Z-tensor with nonnegative diagonal entries is an M-tensor [26], we have B is an M-tensor. Let C = A B. Then c1111 = 4.5, c2222 = 5, c1222 = −1, c2111 = −1, and the other entries are zeros. From Equation (1.1), we obtain {3.7192, 5.7808} is the set of the H-eigenvalues of C. Then τ (C) = 3.7192. By Theorem 3.3(2), we have (C) τ (B) ≤ ττ(A) = 3.7192. Let A be an m-order n-dimensional tensor with m ≥ 2. The determinant of A, denoted by det(A), is the resultant of the homogeneous polynomial system Axm−1 = 0. It is known that eigenvalues of A are roots of det(λI − A) (see [19]). It is complicated to compute the determinant of tensors. Here, we give some properties for the determinant of the Fan product of M-tensors. Theorem 4.4. Let A = (ai1 i2 ···im ) , B = (bi1 i2 ···im ) ∈ R[m,n] be M-tensors. (1) Then
d
|det (A B)| ≥ (τ (A B)) ≥
d min {aii···i τ (B) + bii···i τ (A)} − τ (A)τ (B)
1≤i≤n
d
≥ (τ (A)τ (B)) , n−1
where d = n (m − 1) . (2) If det (A B) = 0. Then det (A) = 0 or det (B) = 0. (3) If det (A B) = 0 and aii···i bii···i = 0 for all i ∈ [n]. Then det (A) = 0 and det (B) = 0. (4) If A is a singular M-tensor and B is a nonsingular M-tensor, and aii···i > 0 for all i ∈ [n]. Then |det (A B)| > 0. n−1
Proof. (1) We know that the number of eigenvalues of A B is d = n (m − 1) and the product of all the eigenvalues is equal to det(A B) (see [19]). Since τ (A B) = d min {Re(λ) : λ ∈ σ(A B)}, we have |det (A B)| ≥ (τ (A B)) . By Theorem 3.3(2), the proof of (1) is completed. (2) Since det (A B) = 0, by (1) and Lemma 2.5(1), we have τ (A) = 0 or τ (B) = 0. Hence, det (A) = 0 or det (B) = 0. (3) Since det (A B) = 0, by (2), we have τ (A) = 0 or τ (B) = 0. Without loss of generality, suppose τ (A) = 0. By (1), we obtain min {aii···i τ (B)} = 0. Since aii···i = 0 1≤i≤n
for all i ∈ [n], we get τ (B) = 0. Hence, det (B) = 0. (4) By Lemma 2.5, we have τ (A) = 0 and τ (B) > 0. Since aii···i > 0 for all i ∈ [n], by Theorem 4.4(1), we have |det (A B)| > 0. 2
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Declaration of Competing Interest The authors have no competing interests for the research conducted in this paper. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 11371109 and No. 11801115), the Natural Science Foundation of Heilongjiang Province (No. QC2018002). References [1] B.D. Anderson, E.I. Jury, Stability of multidimensional digital filters, IEEE Trans. Circuits Syst. 21 (1974) 300–304. [2] B.D. Anderson, N.K. Bose, E.I. Jury, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control 20 (1975) 55–66. [3] N.K. Bose, P.S. Kamat, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process. 22 (1974) 307–314. [4] N.K. Bose, R.W. Newcomb, Tellegon’s theorem and multivariable realizability theory, Int. J. Electron. 36 (1974) 417–425. [5] G. Cheng, New bounds for the minimum eigenvalue of the Fan product of two M-matrices, Czechoslovak Math. J. 64 (2014) 63–68. [6] W. Ding, L. Qi, Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl. 439 (2013) 3264–3278. [7] K. Fan, Inequalities for M-matrices, Indag. Math. 67 (1964) 602–610. [8] M. Fang, Bounds on eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 425 (2007) 7–15. [9] S. Friedland, S. Gaubert, L. Han, Perron-Frobenius theorems for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013) 738–749. [10] M.S. Gowda, Z. Luo, L. Qi, N. Xiu, Z-tensors and complementarity problems, arXiv:1510.07933v2. [11] G.H. Hardy, J.E. Littlewood, G. Polya, Inequality, Cambridge University Press, Cambridge, 1934. [12] J. He, T. Huang, Inequalities for M-tensors, J. Inequal. Appl. 2014 (2014) 114. [13] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. [14] S. Hu, Z. Huang, L. Qi, Strictly nonnegative tensors and nonnegative tensor partition, Sci. China Math. 57 (2014) 181–195. [15] M. Kannan, N. Shaked-Monderer, A. Berman, Some properties of strong H-tensors and general H-tensors, Linear Algebra Appl. 476 (2015) 42–55. [16] M. Kannan, N. Shaked-Monderer, A. Berman, On weakly irreducible nonnegative tensors and interval hull of some classes of tensors, Linear Multilinear Algebra 64 (2016) 667–679. [17] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, in: Proceedings 1st IEEE International Workshop on Computational Advances of Multisensor Adaptive Processing, 2005, pp. 129–132. [18] Q. Ni, L. Qi, F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control 53 (2008) 1096–1107. [19] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302–1324. [20] L. Qi, Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. [21] J. Shao, L. You, On some properties of three different types of triangular blocked tensors, Linear Algebra Appl. 511 (2016) 110–140. [22] L. Sun, B. Zheng, J. Zhou, H. Yan, Some inequalities for the Hadamard product of tensors, Linear Multilinear Algebra 66 (2018) 1199–1214. [23] G. Wang, Y. Wang, Y. Zhang, Some inequalities for the Fan product of M-tensors, J. Inequal. Appl. 2018 (2018) 257. [24] Y. Yang, Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl. 31 (2010) 2517–2530.
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