Let ℝ be the real field. An *m*-th order *n* dimensional square tensor ℬ consists of *n*^{m} entries in ℝ, which is defined as follows:
$$\mathcal{B}=\left({b}_{{i}_{1}{i}_{2}\mathrm{...}{i}_{m}}\right),{b}_{{i}_{1}{i}_{2}\mathrm{...}{i}_{m}}\in \mathbb{R},1\le {i}_{1},{i}_{2},\mathrm{...}{i}_{m}\le n$$

ℬ is called nonnegative if *b*_{i1i2…im} ≥ 0. To an n-vector *x*, real or complex, we define the n-vector:
$$\mathcal{B}{x}^{m-1}={\left({\displaystyle \sum _{}^{n}{b}_{i{i}_{2}\mathrm{...}{i}_{m}}{x}_{{i}_{2}}\mathrm{...}}{x}_{{i}_{m}}\right)}_{1\le i\le n}.$$

and
$${x}^{\left[m-1\right]}={\left({x}_{i}^{m-1}\right)}_{1\le i\le n}.$$

If ℬ*x*^{m−1} = *λx*[^{m−1}, *x* and *λ* are all real, then *λ* is called an H-eigenvalue of ℬ and *x* an H-eigenvector of ℬ associated with *λ*. If ℬ*x*^{m−1} = *λx*, *x* and *λ* are all real, then *λ* is called a Z-eigenvalue of ℬ and *x* a Z-eigenvector of ℬ associated with *λ* [2, 3].

Assume that *p, q, m* and *n* are positive integers, and *m, n* ≥ 2. In this paper we consider a nonnegative (*p, q*)-th order *m* × *n* dimensional rectangular tensor. For any vector *x* and any real number *α*, denote ${x}^{\left[\alpha \right]}={\left[{x}_{1}^{\alpha},\dots ,{x}_{n}^{\alpha}\right]}^{T}$.

Let *Ax*^{p−1} *y*^{q} be a vector in ℝ^{m} such that
$${\left(\mathcal{A}{x}^{p-1}{y}^{q}\right)}_{i}={\displaystyle \sum _{{i}_{2},\dots ,{i}_{p}=1}^{m}{\displaystyle \sum _{{j}_{1},\dots ,{j}_{q}=1}^{n}{a}_{i{i}_{2}\dots {i}_{p}{j}_{1}\dots {j}_{q}}{}_{{}_{}\dots {x}_{{i}_{p}}{x}_{{j}_{1}}\dots {y}_{jq}}}},$$

where *i* = 1,..., *m*. Let *Ax*^{p} *y*^{q−1} be a vector in ℝ^{n} such that
$${\left(\mathcal{A}{x}^{p-1}{y}^{q}\right)}_{i}={\displaystyle \sum _{{i}_{2},\dots ,{i}_{p}=1}^{m}{\displaystyle \sum _{{j}_{1},\dots ,{j}_{q}=1}^{n}{a}_{{i}_{1}{i}_{2}\dots {i}_{p}\text{\hspace{0.17em}}j\text{}{j}_{2}\dots {j}_{q}}{}_{{}_{}\dots {x}_{{i}_{p}}{y}_{{j}_{2}}\dots {y}_{{j}_{q}},}}}$$

where *j* = 1,..., *n*. Throughout this paper, we denote *M = p + q*. If there are a number *λ* ∈ ℂ, vectors *x* ∈ ℂ^{m}\{0}, and *y* ∈ ℂ^{n}\{0} such that
$$\{\begin{array}{c}\mathcal{A}{x}^{p-1}{y}^{q}=\lambda {x}^{\left[M-1\right]}\\ \mathcal{A}{x}^{p}{y}^{q-1}=\lambda {y}^{\left[M-1\right]}\end{array},$$

then *λ* is called the singular value of *A*, and (*x, y*) is the left and right eigenvectors pair of *A*, associated with *λ*. If *λ* ∈ ℝ, *x* ∈ ℝ^{m}, and *y* ∈ ℝ^{n}, then we say that *λ* is an H-singular value of *A*, and (*x,y*) is the left and right H-eigenvectors pair associated with *λ* [1,4].

We denote the cone $\{x\in {\mathbb{R}}^{n}|{x}_{i}\ge \left(resp.>\right)0,i=1,\mathrm{...},n\}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}{\mathbb{R}}_{+}^{n}\left(\text{resp}\text{.}\text{\hspace{0.17em}}{\mathbb{R}}_{++}^{n}\right)$. The Perron-Frobenius Theorem for nonnegative irreducible rectangular tensors was given in [1, 4].

**Theorem 1.1:** *Let A be a* (*p, q*)-*th order m × n dimensional nonnegative tensor. Then λ*_{0}(*A*) *is the largest singular value with nonnegative left and right eigenvectors pair* $\left(x,y\right)\in \text{\hspace{0.17em}}{\mathbb{R}}_{+}^{m}\times \text{\hspace{0.17em}}{\mathbb{R}}_{+}^{n}$ *corresponding to it*.Recently, there are so many results about the properties of square tensor [5–9], especially the upper bounds for the Z-spectral radius and H-spectral radius of a nonnegative square tensor [10–18]. However, there are no results about the upper bounds for the largest singular value of a nonnegative rectangular tensor except the following one [1].

**Theorem 1.2:** *Suppose that A* ≥ 0 *is a* (*p, q*)-*th order m × n dimensional rectangular tensor. Then*
$${\lambda}_{0}\left(\mathcal{A}\right)\le \underset{1\le i\le m,\text{\hspace{0.17em}}1\le j\le n}{\mathrm{max}}\{{r}_{i}\left(\mathcal{A}\right),{c}_{j}\left(\mathcal{A}\right)\},$$where
$$\begin{array}{l}{r}_{i}\left(\mathcal{A}\right)={\displaystyle \sum _{{i}_{2},\dots ,{i}_{p}=1}^{m}{\displaystyle \sum _{{j}_{1},\dots ,{j}_{q}=1}^{n}{a}_{{i}_{1}{i}_{2}\dots {i}_{p}{j}_{1}\dots {j}_{q},}}}\\ {c}_{j}\left(\mathcal{A}\right)={\displaystyle \sum _{{i}_{1},\dots ,{i}_{p}=1}^{m}{\displaystyle \sum _{{j}_{2},\dots ,{j}_{q}=1}^{n}{a}_{{i}_{1}\dots {i}_{p}{j}_{2}\dots {j}_{q}.}}}\end{array}$$Our goal in this paper is to show a tighter upper bound for the largest singular value of a nonnegative rectangular tensor when *m = n*. Let *s*_{i} (*A*) = max{*c*_{i}, (*A*), *r*_{i} (*A*)}, and
$${r}_{i}^{j}\left(\mathcal{A}\right)={r}_{i}\left(\mathcal{A}\right)-{a}_{ij}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{i}^{j}\left(\mathcal{A}\right)={c}_{i}\left(\mathcal{A}\right)-{a}_{j\dots jij\dots j,\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\text{\hspace{0.17em}}i\ne j.$$In particular, we give our main result as follows.

**Theorem 1.3:** *Suppose that A* ≥ 0 *is a* (*p, q*)-*th order n × n dimensional rectangular tensor. Then*
$${\lambda}_{0}\left(\mathcal{A}\right)\le \text{\hspace{0.17em}}\mathrm{max}\text{\hspace{0.17em}}\{{\Theta}_{1}\left(\mathcal{A}\right),{\Theta}_{2}\left(\mathcal{A}\right),{\Theta}_{3}\left(\mathcal{A}\right),{\Theta}_{4}\left(\mathcal{A}\right)\},$$where
$$\begin{array}{l}{\Theta}_{1}\left(\mathcal{A}\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{i,j\in N,j\ne i}{\mathrm{max}}\frac{1}{2}\{{a}_{i\dots i\text{\hspace{0.17em}}i\dots i}+\text{\hspace{0.17em}}{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{r}_{i}^{j}\left(\mathcal{A}\right)+{\Delta}_{1}^{\frac{1}{2}}\left(\mathcal{A}\right)\},\\ {\Theta}_{2}\left(\mathcal{A}\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{i,j\in N,j\ne i}{\mathrm{max}}\frac{1}{2}\{{a}_{i\dots i\text{\hspace{0.17em}}i\dots i}+\text{\hspace{0.17em}}{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{c}_{i}^{j}\left(\mathcal{A}\right)+{\Delta}_{2}^{\frac{1}{2}}\left(\mathcal{A}\right)\},\\ {\Theta}_{3}\left(\mathcal{A}\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{i,j\in N,j\ne i}{\mathrm{max}}\frac{1}{2}\{{a}_{i\dots i\text{\hspace{0.17em}}i\dots i}+\text{\hspace{0.17em}}{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{r}_{i}^{j}\left(\mathcal{A}\right)+{\Delta}_{3}^{\frac{1}{2}}\left(\mathcal{A}\right)\},\\ {\Theta}_{4}\left(\mathcal{A}\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{i,j\in N,j\ne i}{\mathrm{max}}\frac{1}{2}\{{a}_{i\dots i\text{\hspace{0.17em}}i\dots i}+\text{\hspace{0.17em}}{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{c}_{i}^{j}\left(\mathcal{A}\right)+{\Delta}_{4}^{\frac{1}{2}}\left(\mathcal{A}\right)\},\end{array}$$and
$$\begin{array}{l}{\Delta}_{1}\left(\mathcal{A}\right)={\left({a}_{i\dots i\text{\hspace{0.17em}}i\dots i}-{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{r}_{i}^{j}\left(\mathcal{A}\right)\right)}^{2}+4{a}_{ij\dots jj\dots j}{r}_{j}\left(\mathcal{A}\right),\\ {\Delta}_{2}\left(\mathcal{A}\right)={\left({a}_{i\dots i\text{\hspace{0.17em}}i\dots i}-{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{c}_{i}^{j}\left(\mathcal{A}\right)\right)}^{2}+4{a}_{j\dots jij\dots j}{c}_{j}\left(\mathcal{A}\right),\\ {\Delta}_{3}\left(\mathcal{A}\right)={\left({a}_{i\dots i\text{\hspace{0.17em}}i\dots i}-{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{r}_{i}^{j}\left(\mathcal{A}\right)\right)}^{2}+4{a}_{ij\dots jj\dots j}{c}_{j}\left(\mathcal{A}\right),\\ {\Delta}_{4}\left(\mathcal{A}\right)={\left({a}_{i\dots i\text{\hspace{0.17em}}i\dots i}-{a}_{j\dots j\text{\hspace{0.17em}}j\dots j}+{c}_{i}^{j}\left(\mathcal{A}\right)\right)}^{2}+4{a}_{j\dots jij\dots j}{r}_{j}\left(\mathcal{A}\right).\end{array}$$

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