Abstract
An S-type upper bound for the largest singular value of a nonnegative rectangular tensor is given by breaking N = {1, 2, … n} into disjoint subsets S and its complement. It is shown that the new upper bound is smaller than that provided by Yang and Yang (2011). Numerical examples are given to verify the theoretical results.
1 Introduction
Let ℝ(ℂ) be the real (complex) field, p, q, m, n be positive integers, m, n ≥ 2, N = {1, 2,…, n}, and
where ai1…ipj1…jq ∈ ℝ for ik = 1,…, m,; k = 1,… p, and jk = 1,… n, k = 1,… q. When p = q = 1,
For any vector x and any real number α, denote
where i = 1,…, m. Similarly, let
where j = 1,…, n. Let l = p + q. If there are a number λ ∈ ℂ, vectors x ∈ λm}\{0}, and y ∈ ℂn\{0} such that
then λ is called the singular value of
is the largest singular value [2].
Note here that the notion of singular values for tensors was first proposed by Lim in [3]. When l is even, the definition in [1] is the same as in [3]. When l is odd, the definition in [1] is slightly different from that in [3], but parallel to the definition of eigenvalues of square matrices [4]; see [1] for details. For the sake of simplicity, the definition of singular values in this paper is the definition in [1].
We recall the weak Perron-Frobenius theorem for nonnegative rectangular tensors, which was given in [2].
Theorem 1.1 [2, Theorem 2]
Let
The largest singular value of a nonnegative rectangular tensor has a wide range of practical applications in the strong ellipticity condition problem in solid mechanics [5, 6] and the entanglement problem in quantum physics [7, 8]. Recently, there are many results about the properties of square tensors, especially the upper bounds for the ℤ-spectral radius and ℍ-spectral radius of a nonnegative square tensor [9–13]. However, there are no results about the upper bounds for the largest singular value of a nonnegative rectangular tensor except the following one [2].
Theorem 1.2 see [2, Theorem 4]
Let
Throughout this paper, we assume m = n. Our goal in this paper is to give a new upper bound for the largest singular value of a nonnegative rectangular tensor, and prove that the new upper bound is smaller than that in Theorem 1.2.
2 Main results
We begin with some notation. Given a nonempty proper subset S of N, we denote
and then
This implies that for a nonnegative rectangular tensor
where
and
By Theorem 1.2, the following lemma is easily obtained.
Let
Let
where
Let λ0 be the largest singular value of
Let
Then, at least one of xt and xh is nonzero, and at least of yf and yg is nonzero. Next, we present four cases to prove that.
Case I: Suppose that
(i) If xh ≥ xt, then xh = max{wj : i ∈ N}. By the h-th equality in (1), we have
Hence,
If xt = 0, then
i.e.,
From Lemma 2.1, we have λ0-at… tt… t ≥ 0. Multiplying (3) with (4), we have
Note that
Solving (5) gives
(ii) If xt ≥ xh, then xt = max{wj : i ∈ N}. Similarly to the proof of (i), we can obtain that
This gives
Case II: Suppose that ws = yf,
This gives
If yf ≥ yg, then yf = max{wi : i ∈ N}. Similarly to the proof of (ii) in Case I, we have
This gives
Case III: Suppose that ws = xt,
This gives
If xt ≥ yg, then xt = max{wi : i ∈ N}. Similarly to the proof of (ii) in Case I, we can obtain that
This gives
Case IV: Suppose that
This gives
If yf ≥ xh, then yf = max{wi} : i ∈ N}. Similarly to the proof of (ii) in Case I, we can obtain that
This gives
The conclusion follows from Case I, II, III and IV. □
Next, we compare the upper bound in Theorem 2.2 with that in Theorem 1.2.
Let
Here, we only prove
Case I: Suppose that
(i) For any
Hence,
Furthermore,
which implies
(ii) For any
Similarly to the proof of (i), we can obtain
Hence,
Case II: Suppose that
(iii) For any
Similarly to the proof of (i), we can obtain
Hence,
(iv) For any
Similarly to the proof of (ii), we can obtain
Hence,
The conclusion follows from Cases I and II. □
3 Numerical examples
In this section, two numerical examples are given to verify the theoretical results.
Let
LetS ={1, 2}. Obviously
By Theorem 2.2, we have
In fact, λ0 = 31.3781. This example shows that the upper bound in Theorem 2.2 is smaller than that inTheorem 1.2.
Let
otheraijkl = 0. By Theorem 2.2, we have
In fact, λ0 = 3. This example shows that the upper bound in Theorem 2.2 is sharp.
4 Conclusions
In this paper, by breaking N into disjoint subsets S and its complement, an S-type upper bound
Acknowledgement
The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper. This work is supported by the National Natural Science Foundation of China (Nos.11361074,11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066).
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