Let
\(\mathbb{R}\) (respectively,
\(\mathbb{C}\)) be the real (respectively, complex) field. Assume that
\(p,q,m,n\) are positive integers,
\(m,n\geq2\),
\(l=p+q\), and
\(N=\{1,2,\ldots,n\}\). A real
\((p,q)\)th order
\(m\times n\) dimensional rectangular tensor (or simply a real rectangular tensor)
\(\mathcal{A}\) is defined as follows:
$$\mathcal{A}=(a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}),\quad a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}\in\mathbb{R}, 1\leq i_{1},\ldots,i_{p}\leq m, 1\leq j_{1},\ldots ,j_{q}\leq n. $$
A real rectangular tensor
\(\mathcal{A}\) is called nonnegative if
\(a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}\geq0\) for
\(i_{k}=1,\ldots,m, k=1,\ldots, p\), and
\(j_{v}=1,\ldots, n,v=1,\ldots,q\).
For vectors
\(x=(x_{1},\ldots, x_{m})^{\textrm{T}}\),
\(y=(y_{1},\ldots,y_{n})^{\textrm{T}}\) and a real number
α, let
\(x^{[\alpha]}=(x_{1}^{\alpha},x_{2}^{\alpha},\ldots, x_{m}^{\alpha})^{\textrm{T}}\),
\(y^{[\alpha]}=(y_{1}^{\alpha},y_{2}^{\alpha},\ldots,y_{n}^{\alpha})^{\textrm{T}}\),
\(\mathcal{A}x^{p-1}y^{q}\) be an
m dimension real vector whose
ith component is
$$\bigl(\mathcal{A}x^{p-1}y^{q}\bigr)_{i}=\sum _{i_{2},\ldots,i_{p}=1}^{m}\sum_{j_{1},\ldots,j_{q}=1}^{n}a_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}}x_{i_{2}} \cdots x_{i_{p}}y_{j_{1}}\cdots y_{j_{q}}, $$
and
\(\mathcal{A}x^{p}y^{q-1}\) be an
n dimension real vector whose
jth component is
$$\bigl(\mathcal{A}x^{p}y^{q-1}\bigr)_{j}=\sum _{i_{1},\ldots,i_{p}=1}^{m}\sum_{j_{2},\ldots,j_{q}=1}^{n}a_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}}x_{i_{1}} \cdots x_{i_{p}}y_{j_{2}}\cdots y_{j_{q}}. $$
If
\(\lambda\in\mathbb{C}\),
\(x\in\mathbb{C}^{m}\backslash\{0\}\), and
\(y\in \mathbb{C}^{n}\backslash\{0\}\) are solutions of
$$ \textstyle\begin{cases} \mathcal{A}x^{p-1}y^{q}=\lambda x^{[l-1]},\\ \mathcal{A}x^{p}y^{q-1}=\lambda y^{[l-1]}, \end{cases} $$
then we say that
λ is a singular value of
\(\mathcal{A}\),
x and
y are a left and a right eigenvectors of
\(\mathcal{A}\), associated with
λ. If
\(\lambda\in\mathbb{R}, x\in\mathbb{R}^{m}\), and
\(y\in\mathbb{R}^{n}\), then we say that
λ is an H-singular value of
\(\mathcal{A}\),
x and
y are a left and a right H-eigenvectors of
\(\mathcal{A}\), associated with H-singular value
λ [
5]. Here,
$$\lambda_{0}=\max\bigl\{ |\lambda|:\lambda \text{ is a singular value of } \mathcal{A}\bigr\} $$
is called the largest singular value [
6].
The definition of singular values for tensors was first introduced in [
7]. Note here that when
l is even, the definitions in [
5] is the same as in [
7], and when
l is odd, the definition in [
5] is slightly different from that in [
7], but parallel to the definition of eigenvalues of square matrices [
8]; see [
5] for details.
Recently, many people focus on bounding the largest singular value for nonnegative rectangular tensors [
6,
9,
10]. For convenience, we first give some notation. Given a nonempty proper subset
S of
N, we denote
$$\begin{aligned}& \Delta^{N}:=\bigl\{ (i_{2},\ldots, i_{p},j_{1}, \ldots,j_{q}): i_{2},\ldots, i_{p},j_{1}, \ldots ,j_{q}\in N\bigr\} , \\& \Delta^{S}:=\bigl\{ (i_{2},\ldots, i_{p},j_{1}, \ldots,j_{q}): i_{2},\ldots, i_{p},j_{1}, \ldots ,j_{q}\in S\bigr\} , \\& \Omega^{N}:=\bigl\{ (i_{1},\ldots, i_{p},j_{2}, \ldots,j_{q}): i_{1},\ldots, i_{p},j_{2}, \ldots ,j_{q}\in N\bigr\} , \\& \Omega^{S}:=\bigl\{ (i_{1},\ldots, i_{p},j_{2}, \ldots,j_{q}): i_{1},\ldots, i_{p},j_{2}, \ldots ,j_{q}\in S\bigr\} , \end{aligned}$$
and then
$$\overline{\Delta^{S}}=\Delta^{N}\backslash \Delta^{S},\qquad \overline{\Omega ^{S}}=\Omega^{N} \backslash\Omega^{S}. $$
This implies that, for a nonnegative rectangular tensor
\(\mathcal {A}=(a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}})\), we have, for
\(i,j\in S\),
$$\begin{aligned}& r_{i}(\mathcal{A})=\sum_{i_{2},\ldots,i_{p},j_{1},\ldots,j_{q}\in N\atop \delta_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}}=0}a_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}}=r_{i}^{\Delta^{S}}( \mathcal{A})+r_{i}^{\overline{\Delta^{S}}}(\mathcal{A}),\quad r_{i}^{j}( \mathcal{A})=r_{i}(\mathcal{A})-a_{ij\cdots jj\cdots j}, \\& c_{j}(\mathcal{A})=\sum_{i_{1},\ldots,i_{p},j_{2},\ldots,j_{q}\in N\atop \delta_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}}=0}a_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}}=c_{j}^{\Omega^{S}}( \mathcal{A})+c_{j}^{\overline{\Omega^{S}}}(\mathcal{A}),\quad c_{j}^{i}( \mathcal{A})=c_{j}(\mathcal{A})-a_{i\cdots iji\cdots i}, \end{aligned}$$
where
$$\delta_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}= \textstyle\begin{cases} 1,& \text{if }i_{1}=\cdots=i_{p}=j_{1}=\cdots=j_{q},\\ 0,&\text{otherwise}, \end{cases} $$
and
$$\begin{aligned}& r_{i}^{\Delta^{S}}(\mathcal{A})=\sum_{(i_{2},\ldots,i_{p},j_{1},\ldots ,j_{q})\in\Delta^{S}\atop \delta_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}}=0}a_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}},\qquad r_{i}^{\overline{\Delta ^{S}}}( \mathcal{A})=\sum_{(i_{2},\ldots,i_{p},j_{1},\ldots,j_{q})\in \overline{\Delta^{S}}}a_{ii_{2}\cdots i_{p}j_{1}\cdots j_{q}}, \\& c_{j}^{\Omega^{S}}(\mathcal{A})=\sum_{(i_{1},\ldots,i_{p},j_{2},\ldots ,j_{q})\in\Omega^{S}\atop \delta_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}}=0}a_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}},\qquad c_{j}^{\overline{\Omega ^{S}}}( \mathcal{A})=\sum_{(i_{1},\ldots,i_{p},j_{2},\ldots,j_{q})\in \overline{\Omega^{S}}}a_{i_{1}\cdots i_{p}jj_{2}\cdots j_{q}}. \end{aligned}$$