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Bounds for the Z-eigenpair of general nonnegative tensors

Qilong Liu
• School of Mathematics and Statistics, Yunnan University, Kunming, 650091, China
• Email:
/ Yaotang Li
• Corresponding author
• School of Mathematics and Statistics, Yunnan University, Kunming, 650091, China
• Email:
Published Online: 2016-04-06 | DOI: https://doi.org/10.1515/math-2016-0017

Abstract

In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.

MSC 2010: 15A15; 15A69; 65F25

1 Introduction

We start with some preliminaries. First, denote [n] = {1, 2, …, n}. A real mth order n-dimensional tensor 𝓐 = (ai1i2 im) consists of nm real entries:

$ai1i2...im ∈ ℝ$

where ij = 1, 2, …, n for j ∈ [m] [15]. It is obvious that a vector is an order 1 tensor and a matrix is an order 2 tensor. Moreover, a tensor 𝓐 = (ai1...im) is called nonnegative (positive) if each entry is nonnegative (positive). A tensor 𝓐 is said to be symmetric [6, 7] if its entries ai1 i2im are invariant under any permutation of the indices. We shall denote the set of all real mth order n-dimensional tensors by ℝ[m,n] and the set of all nonnegative mth order n-dimensional tensors by ${ℝ}_{+}^{\left[m,n\right]}$. For an n-dimensional vector x = .x1, x2, …, xn), real or complex, we define the n-dimensional vector:

$𝓐xm−1:=(∑i2,...,im∈ [n]ai i2...imxi2...xim)1≤i≤n,$

and the n-dimensional vector:

$x[m−1]:=(xim−1)1≤i≤n⋅$

The following two definitions were first introduced and studied by Qi and Lim [7, 8].

Definition 1.1 ([7, 8]). Let 𝓐 ∈ ℝ[m,n]. A pair (λ, x) ∈ ℂ× (ℂn\{0}) is called an eigenvalue-eigenvector (or simply eigenpair) of A if they satisfy the equation

$𝓐xm−1 = λx[m−1]⋅$(1)

We call (λ, x) an H-eigenpair if they are both real.

Definition 1.2 ([7, 8]). Let 𝓐 ∈ ℝ[m,n]. A pair (λ, x) ∈ ℂ × (ℂn\{0}) is called an E-eigenvalue and E-eigenvector (or simply E-eigenpair) of 𝓐 if they satisfy the equation

${𝓐xm−1=λx, xTx=1.$(2)

We call (λ, x) a Z-eigenpair if they are both real.

The mth degree homogeneous polynomial of n variables f𝓐(x) associated with an mth order n-dimensional tensor 𝓐 = (ai1i2 ... im) ∈ ℝ[m, n] can be represented as

$f𝓐(x)≡𝓐xm:=∑i1,i2,...,im∈[n]ai1i2..imxi1xi2...xim,$

where xm can be regarded as an mth order n-dimensional rank-one tensor with entries xi1xim [2, 5, 9], and 𝓐xm is the inner product of 𝓐 and xm.

Following concept about weakly symmetric of tensors was first introduced and used by Chang, Pearson, and Zhang [6] for studying the properties of Z-eigenvalue of nonnegative tensor.

Definition 1.3 ([6]). A tensor 𝓐 = (ai1i2im) ∈ ℝ[m, n] is called weakly symmetric, if the associated homogenous of polynomial f𝓐(x) satisfy

$∇ f𝓐(x)= m𝓐xm−1, ∀x∈ ℝn.$

and the right-hand side is not identical to zero.

It should be noted for m = 2, symmetric matrices and weakly symmetric matrices are the same. However, it is shown in [6] that a symmetric tensor is necessarily weakly symmetric for m > 2, but the converse is not true in general. Thus, the results of this paper derived for weakly symmetric tensors, apply also for symmetric tensors. In [8], the notion of irreducible tensors was introduced.

Definition 1.4 ([8]). A tensor 𝓐 = (ai1i2im) ∈ ℝ[m, n] is called reducible if there exists a nonempty proper index subset I ⊂ [n], such that

$ai1i2...im=0, ∀i1∈ I, ∀i2,..., im∉ I,$

otherwise, we say 𝓐 is irreducible.

The Z-spectral radius of a tensor is defined as follows in [10].

Definition 1.5 ([10]). Let 𝓐 ∈ ℝ[m, n]. We define the Z-spectrum of 𝓐, denoted σ(𝓐) to be the set of all Z-eigenvalues of 𝓐. Assume σ(𝓐) ≠ ∅, then the Z-spectral radius of 𝓐, denoted ϱ(𝓐), is defined as

$ϱ(𝓐):= max{|λ|:λ∈σ(𝓐)}.$

Chang, Pearson and Zhang [10] studied the Z-eigenpair problem for nonnegative tensors and presented the following Perron-Frobenius type theorem.

Lemma 1.6 ([10]). If $𝓐\in {ℝ}_{+}^{\left[m,n\right]}$, then there exists a Z-eigenvalue λ0 ≥ 0 and a nonnegative Z-eigenvector x0 ≠ 0 of 𝓐 such that $𝓐{x}_{0}^{m-1}={\lambda }_{0}{x}_{0}$, in particular, if 𝓐 is irreducible, then the eigenvalue x0 and the eigenvector x0 are positive. Furthermore, if $𝓐\in {ℝ}_{+}^{\left[m,n\right]}$ is weakly symmetric irreducible, then the spectral radius ϱ(𝓐) positive Z-eigenvalue with a positive Z-eigenvector.

Z-eigenvalues play a fundamental role in the symmetric best rank-one approximation which has numerous applications in engineering and higher order statistics, such as Statistical Data Analysis [2, 5, 9]. The symmetric best rank-one approximation of 𝓐 = (ai1i2…im) is a rank-one tensor νxm = (νxi1 xi2 … xim), where ν ∈ ℝ, x ∈ ℝn, ||x||2 = 1 and ||x||2 is the Euclidean norm of x in ℝn, such that the Frobenius norm ||𝓐 – νxm||F is minimized. The Frobenius norm of the tensor 𝓐 = (ai1i2 im) has the form

$||𝓐||F:=∑i1,i2,...im∈[n]ai1i2...im2.$

According to [11], νxm is a symmetric best rank-one approximation of 𝓐 if and only if ν is a Z-eigenvalue of 𝓐 with the largest absolute value, while x is a Z-eigenvector of 𝓐 associated with the Z-eigenvalue ν. In particular, when $𝓐\in {ℝ}_{+}^{\left[m,n\right]}$ is weakly symmetric irreducible, $\rho \left(𝓐\right){x}_{0}^{m}$ is a symmetric best rank-one approximation of 𝓐, where x0 is a Z-eigenvector of 𝓐 associated with Z-spectral radius ϱ(𝓐), i.e.,

$minν∈R,x∈ℝn,||x||2=1||𝓐−νxm||F=||𝓐−ϱ(𝓐)x0m||F=||𝓐||F2−ϱ(𝓐)2.$(3)

Thus, we obtain the quotient of the residual of a symmetric best rank-one approximation of tensor 𝓐 and the Frobenius norm of tensor 𝓐 as follows:

$||𝓐−ϱ(𝓐)x0m||F||𝓐||F=1−ϱ(𝓐)2||𝓐||F2.$(4)

By Equalities (3) and (4), if we give a bound of Z-spectral radius of A, then a bound of $\underset{\nu \in ℝ,x\in ℝ,||x|{|}_{2}=1}{\mathrm{min}}||𝓐-\nu {x}^{m}|{|}_{F}$ and $\frac{||𝓐-\rho \left(𝓐\right){x}_{0}^{m}|{|}_{F}}{||𝓐|{|}_{F}}$ will be obtained. It follows from [12-16] that the bound of $\frac{||𝓐-\rho \left(𝓐\right){x}_{0}^{m}|{|}_{F}}{||𝓐|{|}_{F}}$ gives a convergence rate for the greedy rank-one update algorithm.

Recently, some H-spectral of matrices have been successfully extended to higher order tensors [17-19]. For the Z-eigenpair case, Chang, Pearson and Zhang [10] discussed the variation principles of Z-eigenvalues of nonnegative tensors, as a corollary of the main results, they presented a lower bound of the Z-spectral radius for weakly symmetric nonnegative irreducible tensors as follows:

$max{maxi ∈[n]ai i...i,(1n)m−2 mini ∈ [n] ∑i2,...,im∈[n]ai i2...im }≤ϱ(𝓐).$(5)

For a general tensor case, they also provided an upper bound for the Z-spectral radius:

$ϱ(𝓐) ≤ n maxi ∈ [n] ∑i2,...,im∈[n]|ai i2...im|.$(6)

Song and Qi [20] obtained the following upper bound for a general mth order n-dimensional tensor:

$ϱ(𝓐) ≤ maxi ∈ [n] ∑i2,...,im∈[n]|ai i2...im|.$(7)

He and Huang [21] gave a bound for a weakly symmetric positive tensor:

$ϱ(𝓐) ≤ R(𝓐)−l(𝓐)(1−θ(𝓐)),$(8)

where ${r}_{i}\left(𝓐\right)=\sum _{{i}_{2},...,{i}_{m}\in \left[n\right]}{a}_{i{i}_{2}...{i}_{m}}$, for all i ∈ [n],

$RA=maxi∈nriA,rA=mini∈nriAlA= mini1,...,im∈nai1i2...im,andθA=rARA1m.$(9)

Since θ(𝓐) ≤ 1, it is easy to see that the bound (8) is smaller than those in (6) and (7) if the tensor is weakly symmetric positive. Recently, Li, Liu and Vong [22] have given a lower bound and an upper bound for a weakly symmetric nonnegative irreducible tensor:

$dm,n≤ϱ(𝓐) ≤ maxi,j∈[n]{ri(𝓐)+aij...j(δ(𝓐)−m−1m−1)}$(10)

where

$δ(𝓐)=mini,j ∈[n]aij...jr(𝓐)−mini,j ∈[n]aij...j(γ(𝓐)m−1m−γ(𝓐)1m)+γ(𝓐),$(11)$γ(𝓐)=R(𝓐)−mini,j ∈[n]aij...jr(𝓐)−mini,j ∈[n]aij...j,$(12)

and

$dm,n=maxk∈[m]\{1} mini1∈[n][(δ(𝓐)1m−1)minit,t ∈[m]\{1}ai1i2...ik−1ikik+1...im+ minit,t ∈[m]\{1,k}∑ik∈[n]ai1i2...ik−1ikik+1...im].$(13)

They also proved that the upper bound (10) is smaller than that in (8). Furthermore, Li, Liu and Vong [22] obtained the following upper bound for a general mth order n-dimensional tensor:

$ϱA≤mink∈[m]maxik∈[n]⁡∑it=1,t∈[m]∖{k}n|ai1...ik...im|.$(14)

In this paper, we continue this research on the Z-eigenpair and present some bounds as follows: for a weakly symmetric nonnegative irreducible tensor, we present a bound for Z-spectral radius, which improves the bound in (10). For a weakly symmetric nonnegative tensor, we give an upper bound for Z-spectral radius. Furthermore, for a nonnegative tensor and a general tensor, an upper bound for Z-spectral radius is also provided, which is tighter than the bound in (14) in some sense.

Our paper is organized as follows. In Section 2, an upper bound for the ratio of the largest and smallest values of a Z-eigenvector is given. Also, a lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Moreover, an upper bound for the Z-spectral radius of a weakly symmetric nonnegative tensor is provided. An upper bound for the Z-spectral radius of a nonnegative tensor and an upper bound for the Z-spectral radius of a general tensor are obtained in Section 3. Numerical examples are presented in the final section.

We first add a comment on the notation that is used. For a tensor 𝓐, let |𝓐| denote the tensor whose .(i1, … im)-th entry is defined as |ai1 … im|. For a set S, |S| denotes the number of elements of S. The function └x┘ indicates the integer round-down of x. Denote

$Δj;k=⋃S⊆{2,...,m},|S|=k{(i2,...,im):it=j,forallt∈Sandit≠j,forallt∉S}$

where j ∈ [n], k = 0, 1, …, m – 1.

2 Bounds for weakly symmetric nonnegative tensors

In this section, a lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are provided, which improves the bound (10). We first establish a lemma to estimate the ratio of the largest and smallest values of a Z-eigenvector.

Lemma 2.1. Suppose that $𝓐=\left({a}_{{i}_{1}{i}_{2}...{i}_{m}}\right)\in {ℝ}_{+}^{\left[m,n\right]}$ is an irreducible tensor. Then for any Z-eigenpair (λ0, x) of 𝓐 with a positive eigenvector x, we have

$xmaxxmin≥τ(𝓐)1m,$(15)

where ${x}_{min}=\underset{i\in \left[n\right]}{\mathrm{min}}{x}_{i},{x}_{max}=\underset{i\in \left[n\right]}{\mathrm{max}}{x}_{i}$,

$τ(𝓐)=∑k=0⌊m2⌋−1(km−1)(n−1)kβk(𝓐)(α(𝓐)m−k−1m−α(𝓐)k+1m)r(𝓐)−∑k=0⌊m2⌋−1(km−1)(n−1)kβk(𝓐)+α(𝓐),βk(𝓐)=mini,j∈[n]{ai i2,...,:(i2,...,im)∈Δ(j:m−k−1)},α(𝓐)=R(𝓐)−∑k=0⌊m2⌋−1(km−1)(n−1)kβk(𝓐)r(𝓐)−∑k=0⌊m2⌋−1(km−1)(n−1)kβk(𝓐)$(16)

Proof. Since (λ0, x) is a Z-eigenpair of 𝓐 with x being positive, then

${𝓐xm−1=λ0x, xTx=1.$(17)

For simplicity, let xl = xmax, xs = xmin, ${r}_{p}\left(𝓐\right)=\underset{i\in \left[n\right]}{\mathrm{max}}{r}_{i}\left(𝓐\right)=R\left(𝓐\right),{r}_{q}\left(𝓐\right)=\underset{i\in \left[n\right]}{\mathrm{min}}{r}_{i}\left(𝓐\right)=r\left(𝓐\right)$. Consider the ith equation of (17), we obtain

$λ0xi=∑(i2,..,im)∈[n]aii2...imxim≥ail...lx1m−1+∑(i2,..,im)∈Δ(l;m−2)aii2...imx1m−2xs+ ∑(i2,..,im)∈Δ(l;m−3)aii2...imx1m−3xs2+...+∑(i2,..,im)∈Δ(l;m−⌊m2⌋)aii2...imx1m−⌊m2⌋xs⌊m2⌋−1+ ∑(i2,..,im)∈ ∪k=0m−⌊m2⌋ Δ(l;k)aii2...imxsm−1=ail...l(x1m−1−xsm−1)+∑(i2,..,im)∈Δ(l;m−2)aii2...im(x1m−2xs−xsm−1) ∑(i2,..,im)∈Δ(l;m−3)aii2...im(x1m−2xs2−xsm−1)+ ∑(i2,..,im)∈Δ(l;m⌊m2⌋)aii2...im(x1m−⌊m2⌋xs⌊m2⌋−1−xsm−1) + ri(𝓐)xsm−1∑k=0⌊m2⌋−1∑(i2,..,im)∈Δ(l;m−k−1)aii2...im(x1m−k−1xsk−xsm−1)+ri(𝓐)xsm−1.$(18)

Taking i = p in (18) and multiplying by ${x}_{p}^{-1}$ on the both sides of (18) gives

$λ0≥∑k=0⌊m2⌋−1∑(i2,..,im)∈Δ(l;m−k−1)ai i2...im(x1m−k−1xsk−xsm−1xp) +R(𝓐)xsm−1xp≥∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)(x1m−k−1xsk−xsm−1x1) +R(𝓐)xsm−1x1=∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)x1m−k−1xsk+ (R(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))xsm−1x1.$(19)

Similarly, we have

$λ0xi≤ ∑k=0⌊m2⌋−1∑(i2,..,im)∈Δ(l;m−k−1)ai i2...im(xsm−k−1x1k−x1m−1)+ri(𝓐)x1m−1.$(20)

Taking i = q. By (20) and the similar technique to (19) we have

$λ0≤ ∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)x1m−k−1x1k+ (r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))xlm−1xs.$(21)

Combining (19) with (21) together gives

$∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)x1m−k−1xsk+ (R(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))xsm−1x1.≤ ∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)x1m−k−1x1k+ (r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))xlm−1xs.$

Multiplying $\frac{{x}_{l}}{{x}_{s}^{m-1}}$ on the both sides of the above inequality yields

$∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(x1xs)m−k−1+ R(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)≤ ∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(x1xs)k+1 + (r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))(xlxs)m.$(22)

Note that ${\left(\frac{{x}_{l}}{{x}_{s}}\right)}^{m-k-1}\ge {\left(\frac{{x}_{l}}{{x}_{s}}\right)}^{k+1}$, for all $k=0,1,\dots ,⌊\frac{m}{2}⌋-1$. Hence, by (22), we have

$(xlxs)m≥R(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)(r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)):= α(𝓐),$

i.e. $\frac{{x}_{l}}{{x}_{s}}\ge \alpha {\left(𝓐\right)}^{\frac{1}{m}}$, which together with (22) yields

$(xlxs)m≥∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)(α(𝓐)m−k−1m−α(𝓐)k+1m(r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐))+α(𝓐) = τ(𝓐).$

So the desired conclusion follows. □

We next compare the bounds (15) in Lemma 2.1 with the corresponding bounds in Theorem 3.1 of [22], in which the authors presented the following bounds:

$xmaxxmin≥δ(𝓐)1m,$(23)

where δ(𝓐) given as (11).

Lemma 2.2. Suppose that 𝓐 = (ai1i2 …im) ∈ ℝ[m, n] is an irreducible tensor. Then

$τ(𝓐) ≥ α(𝓐) ≥ γ(𝓐) ≥1,$(24)

and

$τ(𝓐) ≥ δ(𝓐) ≥ γ(𝓐) ≥1.$(25)

Proof. It follows from Inequality (3.10) of Li, Liu and Vong [22] that δ(𝓐) ≥ y(𝓐) ≥ 1. Hence, we only prove

$τ(𝓐) ≥ α(𝓐) ≥ γ(𝓐)$(26)

and

$τ(𝓐) ≥ δ(𝓐)$(27)

We have from Equality (16) that

$α(𝓐)=R(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)≥R(𝓐)−β0(𝓐)r(𝓐)−β0(𝓐)=R(𝓐)−mini,j∈[n]aij...jr(𝓐)−mini,j∈[n]aij...j=γ(𝓐).$(28)

Note that $\alpha {\left(𝓐\right)}^{\frac{m-k-1}{m}}\ge \alpha {\left(𝓐\right)}^{\frac{k+1}{m}}$, for all $k=0,1,\dots ,⌊\frac{m}{2}⌋-1$, and

$r(𝓐)>∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐),$

hence

$τ(𝓐)≥α(𝓐)$

which together with (28), yields Inequality (26).

From Equality (16), we obtain

$τ(𝓐)=∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)(α(𝓐)m−1m−α(𝓐)k+1m)r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)+α(𝓐)≥β0(𝓐)(α(𝓐)m−1m−α(𝓐)1m)r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)+α(𝓐)=mini,j∈[n]aij...j(α(𝓐)m−1m−α(𝓐)1m)r(𝓐)−∑k=0⌊m2⌋−1(m−1k)(n−1)kβk(𝓐)+α(𝓐).$(29)

Since α(𝓐). ≥ γ(𝓐) ≥ 1 and m ≥ 2,

$α(𝓐)m−1m−α(𝓐)1m≥γ(𝓐)m−1m−γ(𝓐)1m,$

which together with Inequality (29), yields

$τ(𝓐)≥mini,j∈[n]aij...jr(𝓐)−mini,j∈[n]aij...j(γ(𝓐)m−1m−γ(𝓐)1m)+γ(𝓐)=δ(𝓐).$

which implies Inequality (27) holds. The proof is completed. □

Remark 2.3. If 𝓐 is a nonnegative irreducible tensor, Inequality (25) implies the bounds (15) are always larger than the bounds (23).

Now we establish an upper and a lower bound for Z-spectral radius of a weakly symmetric nonnegative irreducible tensor.

Theorem 2.4. Suppose that $𝓐=\left({a}_{{i}_{1}{i}_{2}...{i}_{m}}\right)\in {ℝ}_{+}^{\left[m,n\right]}$ is an irreducible weakly symmetric tensor. Then

$μ(𝓐) ≤ϱ(𝓐)≤η(𝓐),$(30)

where

$ηA=maxi,j∈[n]∑k=0m−2∑(i2,..,im)∈Δ(l;m−k−1)aii2...imτA−m−k−1m−1+riA,μA=maxk∈[m]∖{1}mini1∈[n]τA1m−1minit,t∈[m]∖{1}ai1...ik−1ikik+1...im+minit,t∈[m]∖{1,k}⁡∑ik∈[n]ai1...ik−1 ikik+1...im,$(31)

and τ(𝓐) is given by Lemma 2.1.

Proof. It follows from Lemma 1.6 that there exists a positive Z-eigenvector x corresponding to ϱ(𝓐). Taking the similar technique of (18) we obtain

$ϱ(𝓐)xi≤∑k=0m−2∑(i2,..,im)∈Δ(s;m−k−1)ai i2...im(xsm−kxlk−1−xlm−1)+ri(𝓐)xlm−1,$(32)

Since xTx = 1, we have ${x}_{i}^{m-1}\le {x}_{i}$, which together with (32) yields

$ϱ(𝓐)xim−1≤∑k=0m−2∑(i2,..,im)∈Δ(s;m−k−1)ai i2...im(xsm−kxlk−1−xlm−1)+ri(𝓐)xlm−1.$

Taking i = l and multiplying ${x}_{l}^{1-m}$ on the both sides of the above inequality gives

$ϱ(A)≤∑k=0m−2∑(i2,..,im)∈Δ(s;m−k−1)aii2...im(xsm−k−1xlk−xlm−1xlm−1)+rl(A)=∑k=0m−2∑(i2,..,im)∈Δ(s;m−k−1)aii2...im(τ(A)−m−k−1m−1)+rl(A)≤∑k=0m−2∑(i2,..,im)∈Δ(s;m−k−1)aii2...im(τ(A)−m−k−1m−1)+rl(A)≤maxi,j∈[n]⁡{∑k=0m−2∑(i2,..,im)∈Δ(j;m−k−1)aii2...im(τ(A)−m−k−1m−1)+ri(A)}.$

This proves the second Inequality (30). On the other hand, it is known from Theorem 3.3 of Li, Liu and Vong [22] that for a weakly symmetric nonnegative irreducible tensor 𝓐, we have

$ϱ(𝓐)≥(xlxs−1) minit,t∈[m]\{1}asi2...ik−1lik+1...im+minit,t∈[m]\{1,k}∑ik∈[n]asi2...ik−1lik+1...im,$

which together with (15) yields

$ϱ(𝓐)≥(τ(𝓐)1m−1) minit,t∈[m]\{1}asi2...ik−1lik+1...im+minit,t∈[m]\{1,k}∑ik∈[n]asi2...ik−1lik+1...im≥mini1∈[n][(τ(𝓐)1m−1) minit,t∈[m]\{1}asi2...ik−1lik+1...im+minit,t∈[m]\{1,k}∑ik∈[n]asi2...ik−1lik+1...im],$

which proves the first Inequality of (30). This proves the theorem. □

Remark 2.5. It follows from Inequality (25) that for the parameters η(𝓐) and μ(𝓐) given by (31) we have

$η(𝓐)≤ maxi,j∈[n]{ri(𝓐)+aij...j(δ(𝓐)−m−1m−1)},$

and

$μ(𝓐)≥dm,n$

This implies the bounds (30) are always better than the corresponding bounds (10).

Remark 2.6. For an irreducible weakly symmetric tensor $𝓐=\left({a}_{{i}_{1}{i}_{2}...{i}_{m}}\right)\in {ℝ}_{+}^{\left[m,n\right]}$, when m and n are very large, the bounds in (30) need more computations than the bounds in (10). As stated in Section 1, by the bounds in (30), we can obtain a more sharp bound of $\underset{\nu \in ℝ,x\in ℝ,||x|{|}_{2}=1}{\mathrm{min}}||𝓐-\nu {x}^{m}|{|}_{F}$, which plays an important role in the symmetric best rank-one approximation [12-16]. This can be seen in the following example.

Example 2.7. Let $𝓐=\left({a}_{{i}_{1}{i}_{2}{i}_{3}{i}_{4}}\right)\in {ℝ}_{+}^{\left[4,2\right]}$ with entries defined as follows:

$a1111 = 10; a2222 = 20; a1122 = a1212 = a2112 = a2121 = 0.25, a1221 = a2211 = 0.3and ai1i2i3i4=0.4, elsewhere.$

It is not difficult see that 𝓐 is a weakly symmetric nonnegative irreducible tensor, and

$||𝓐||F= 22.3989.$

By the bound (10), we have

$0:6914 ≤ ϱ(𝓐) 22.2525,$

which implies

$2.5569 ≤ minν∈ℝ,x∈ℝn,||x||2=1||𝓐−νxm||F=||𝓐||F2−ϱ(𝓐)2≤22.3882.$

By the bound (30), we have

$0.6937 ≤ ϱ(𝓐)≤21.8479,$

which means

$4397374 ≤ minν∈ℝ,x∈ℝn,||x||2=1||𝓐−νxm||F=||𝓐||F2−(𝓐)2≤22.3881.$

Before generalizing the upper bound (30) of Theorem 2.4 to weakly symmetric nonnegative tensor, which will be used in Section 4, we first give a lemma in [20].

Lemma 2.8 ([20]). Suppose that $𝓐\in {ℝ}_{+}^{\left[m,n\right]}$ is weakly symmetric. If 0 ≤ 𝓐 ≤ 𝓑, then ϱ(𝓐) ≤ ϱ(𝓑).

Theorem 2.9. Suppose that $𝓐\in {ℝ}_{+}^{\left[m,n\right]}$ is weakly symmetric. Then

$ϱ(𝓐)≤η(𝓐).$(33)

where η(𝓐) given by Theorem 2.4.

Proof. If 𝓐 is irreducible. Inequality (33) follows from Theorem 2.4. If 𝓐 is reducible. Let ${𝓐}_{t}=𝓐+\frac{1}{t}\epsilon$, where t = 1, 2, … and Ɛ is a tensor with all entries being 1. Then {𝓐t} g is a sequence of weakly symmetric and positive tensors satisfying 0 ≤ 𝓐 < 𝓐t+1 < 𝓐t. By Lemma 2.8, ϱ(𝓐t) is a monotone decreasing sequence with lower bound ϱ(𝓐) so that ϱ(𝓐t) has a limit. Thus, by Theorem 2.4, we have

$ϱ(A)≤ϱ(At)≤maxi,j∈[n]⁡{∑k=0m−2∑(i2,..,im)∈Δ(j;m−k−1)(ai i2...im+1t)(τ(At)−m−k−1m−1)+ri(A)+nm−1t},$

letting t → ∞, note that τ(𝓐t) → τ(𝓐), we obtain ϱ(𝓐) ≤ η(𝓐). This yields the desired conclusion. □

3 Upper bound for nonnegative tensors

In this section, we present an upper bound for the Z-eigenvalue of a nonnegative tensor and an upper bound for the Z-eigenvalue of a general tensor, which improves the bound (14). The following two lemmas will be used.

Lemma 3.1 ([6, 18]). Let 𝓐 = (ai1i2 ... im) ∈ ℝ [m, n] and 𝓐 = (bi1i2 ... im) ∈ ℝ[m, n] be given as follows:

$bi1i2...im=∑π∈Symmaiπ(1)iπ(2)...iπ(m),$(34)

then 𝓑 is symmetric and

$𝓐xm=1m!𝓑xm,$(35)

where Symm is the set of all permutation in [m].

Lemma 3.2 ([10]). If 𝓐 ∈ ℝ[m,n] is weakly symmetric, then σ(𝓐) consists precisely of all critical values of f𝓐(x) = 𝓐xm on Sn–1, where Sn–1 is the standard unit sphere inn.

Based on Lemma 3.2, we have the following lemma.

Lemma 3.3. Suppose that 𝓐 is weakly symmetric. Then

$ϱ(𝓐)=max{|𝓐xm|:xTx=1,x∈ℝn}.$(36)

Proof. Assume that 𝓐 is a weakly symmetric tensor. By Lemma 3.2, we have

$max{|𝓐xm|:xTx=1,x∈ℝn}≤max{|λ|:λ∈σ(𝓐)}ϱ(𝓐).$(37)

On the other hand, assume that λ is a Z-eigenvalue of 𝓐 with Z-eigenvector x. It follows from Equation (1.2) that λ = 𝓐xm, thus

$ϱA=max|λ|:Axm−1=λx,xTx=1,x∈Rn=max|Axm|:Axm−1=λx,xTx=1,x∈Rn≤max|Axm|:xTx=1,x∈Rn,$

which together with (37), yields (36). The proof is completed. □

In the next theorem we have given an upper bound for the Z-eigenvalues of a nonnegative mth order n-dimensional tensor.

Theorem 3.4. Let $𝓐=\left({a}_{{i}_{1}{i}_{2}...{i}_{m}}\right)\in {ℝ}_{+}^{\left[m,n\right]}$. Then for any Z-eigenvalue λ, we have

$|λ| ≤1m!ϱ(|𝓑|)≤1m!η(|𝓑|),$

where 𝓑 given by (34) and η(𝓑) defined as Theorem 2.4.

Proof. Assume that λ is a Z-eigenvalue of 𝓐 with Z-eigenvector x. It follows from Equation (1.2) that λ = 𝓐xm, thus

$|λ|=|Axm|:Axm−1=λx,xTx=1,x∈Rn≤max|Axm|:xTx=1,x∈Rn=1m!max|Bxm|:xTx=1,x∈Rn,$(38)

Note that 𝓐 is a symmetric tensor, by Lemma 3.3, we have

$max{|𝓑xm|:xTx=1,x∈ℝn}=ϱ(𝓑),$

which together with Inequality (38), yields

$|λ| ≤1m!ϱ(𝓑).$(39)

From 𝓐 being nonnegative it follows that 𝓑 is also nonnegative, furthermore, 𝓑 is symmetric. By Theorem 2.9, we obtain

$ϱ(𝓑)≤η(𝓑),$

which together with Inequality (39), yields

$|λ| ≤1m!ϱ(𝓑)≤1m!η(𝓑),$

The proof is completed. □

The following result gives an upper bound for the Z-eigenvalue of a general mth order n-dimensional tensor.

Theorem 3.5. Let 𝓐 = (ai1i2im) ∈ ℝ[m, n] Then for any Z-eigenvalue λ, we have

$|λ| ≤1m!ϱ(|𝓑|)≤1m!η(|𝓑|),$(40)

where 𝓑 given by (34), $|𝓑|=\left(|{b}_{{i}_{1}{i}_{2}...{i}_{m}}|\right)\in {ℝ}_{+}^{\left[m,n\right]}$ and η(|𝓑|) defined as Theorem 2.4.

Proof. Assume that λ is a Z-eigenvalue of 𝓐 with Z-eigenvector x. It follows from Equation (1.2) that λ = 𝓐xm, thus

$|λ|=|Axm|:Axm−1=λx,xTx=1,x∈Rm≤max|Axm|:xTx=1,x∈Rn=1m!max|Bxm|:xTx=1,x∈Rn≤1m!max|Bxm|:xTx=1,x∈Rn≤1m!max|Bxm|:xTx=1,x∈Rn.$(41)

Since |𝓑| is a symmetric tensor, by Lemma 3.3, we obtain

$max{||𝓑|xm|:xTx=1,x∈ℝn}=ϱ(𝓑),$

which together with Inequality (41), yields

$|λ|≤1m!(|𝓑|).$(42)

Obviously, |𝓑| is a nonnegative symmetric tensor, by Theorem 2.9, we have

$ϱ(|𝓑|)≤η(|𝓑|),$

which together with Inequality (42), yields

$|λ|≤1m!ϱ(|𝓑|) ≤1m!η(|𝓑|).$

The proof is completed. □

By means of the proof technique of Theorem 3.5, the following conclusions are obtained.

Corollary 3.6. Suppose that 𝓐 = (ai1i2im) ∈ ℝ[m, n] is a symmetric tensor. Then for any Z-eigenvalue λ, we have

$λ≤ϱ(|𝓐|)≤η(|𝓐|),$

where |𝓐| = (|ai1i2im|) ∈ ℝ[m, n], and η(|𝓐|) defined as Theorem 2.4.

Proof. Assume that λ is a Z-eigenvalue of 𝓐 with Z-eigenvector x. It follows from Equation (1.2) that λ = 𝓐xm, thus

$|λ|=|Axm|:Axm−1=λx,xTx=1,x∈Rn≤max|Axm|:,xTx=1,x∈Rn≤max|A||x|m|:xTx=1,x∈Rn=max||A|xm|:,xTx=1,x∈Rn=ϱ|A|≤η|A|.$

The proof is completed. □

4 Comparisons with existing bounds

In this section, we will give some comparisons between our bounds and existing bounds for the Z-spectral radius for tensors. For a weakly symmetric nonnegative irreducible tensor, from Lemma 2.2, we obtain the bounds in (30), which are always better than the ones in (10). In particular, for a weakly symmetric positive tensor, the upper bound in (30) is always smaller than ones in (8).

The following example given in [10, 22] shows the efficiency of the new bound (30).

Example 4.1. Let 𝓐 ∈ ℝ[4, 2] be a symmetric tensor defined by

$a1111=12,a2222=3, and ai1...i4=13elsewhere.$

It follows from [10] that

$ϱ(𝓐) ≈ 3.1092.$

By the bound (7), we have

$ϱ(𝓐) ≤ 5.3333.$

By the bound (8), we have

$ϱ(𝓐) ≤ 5.2846.$

By the bound (10), we have

$0.7330 ≤ ϱ(𝓐) ≤ 5.1935.$

By the bound (30), we have

$0.7663 ≤ ϱ(𝓐) ≤ 4.5147.$

This example shows that the bound (30) is the best.

For a general tensor, the following example shows that our bound in (40) is better than the bound in (14) for some tensors.

Example 4.2. Randomly generate 1000 tensors of 4th order 3-dimensional tensor such that the elements of each tensor are generated by uniform distribution (–0.05, 0.40). We compare the upper bounds of the Z-spectral radius of general tensor in (14) and (40). The numerical results are showed in Fig. 1. The x-axis of Fig. 1 refers to the sth random generated tensor. The plus symbol in blue color denotes the difference between the upper bound (14) and (40). As observed from Fig. 1, there are 96.2% of cases above the x-axis.

Fig. 1

The randomly generated results for Example 4.2

Acknowledgement

The authors are grateful to the anonymous referees for their valuable comments, which helped improve the quality of the paper. The work is supported by National Natural Science Foundations of China (11361074), Natural Science Foundations of Yunnan Province (2013FD002) and IRTSTYN.

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Accepted: 2016-03-07

Published Online: 2016-04-06

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation