1 Introduction

A fourth-order real tensor \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in \mathbb{R}^{m\times n\times m\times n}\) is called partially symmetric [1] if it has the following symmetry:

$$a_{i_{1}i_{2}i_{3}i_{4}}=a_{i_{3}i_{2}i_{1}i_{4}}=a_{i_{1}i_{4}i_{3}i_{2}}, \quad\forall i_{1}, i_{3}\in[m], \forall i_{2}, i_{4}\in[n], $$

where \([m]=\{1,2,\ldots,m\}\) and \([n]=\{1,2,\ldots,n\}\). Such a tensor often arises in nonlinear elastic materials analysis [2, 3] and entanglement studies in quantum physics [4–6]. For this tensor, there are many kinds of eigenvalues such as H-eigenvalues, Z-eigenvalues, and D-eigenvalues [7, 8]; here we only discuss its M-eigenvalues [1, 9].

Definition 1

([9])

Let \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{m\times n\times m\times n}\) be a partially symmetric tensor, and let \(\lambda \in\mathbb{R}\). Suppose that there are real vectors \(x\in\mathbb {R}^{m}\) and \(y\in\mathbb{R}^{n}\) such that

$$ \left \{ \textstyle\begin{array}{l} \mathcal{A}\cdot yxy=\lambda x,\\ \mathcal{A}xyx\cdot=\lambda y,\\ x^{T}x=1,\\ y^{T}y=1, \end{array}\displaystyle \right . $$
(1)

where \(\mathcal{A}\cdot yxy\in\mathbb{R}^{m}\) and \(\mathcal{A}xyx\cdot \in\mathbb{R}^{n}\) with ith components

$$ (\mathcal{A}\cdot yxy)_{i}=\sum_{i_{3}\in[m]}\sum _{i_{2},i_{4}\in [n]}a_{ii_{2}i_{3}i_{4}}y_{i_{2}}x_{i_{3}}y_{i_{4}} $$

and

$$ (\mathcal{A}xyx\cdot)_{i}=\sum_{i_{1},i_{3}\in[m]}\sum _{i_{2}\in [n]}a_{i_{1}i_{2}i_{3}i}x_{i_{1}}y_{i_{2}}x_{i_{3}}. $$

Then λ is called an M-eigenvalue of \(\mathcal{A}\) with left M-eigenvector x and right M-eigenvector y.

Note that M-eigenvalues of a fourth-order partially symmetric tensor always exist [1]. They have a close relation to many problems in the theory of elasticity and quantum physics [1, 9, 10]. For example, the largest M-eigenvalue of \(\mathcal {A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{m\times n\times m\times n}\), denoted by

$$ \lambda^{\star}=\max\{\lambda: \lambda \text{ is an M-eigenvalue of } \mathcal{A}\}, $$

is the optimum solution of the problem (for details, see [9])

$$ \begin{gathered} \text{max} \quad f(x, y)=\sum _{i_{1},i_{3}=1}^{m}\sum_{i_{2},i_{4}=1}^{n}a_{i_{1}i_{2}i_{3}i_{4}}x_{i_{1}}y_{i_{2}}x_{i_{3}}y_{i_{4}}, \\ \quad\text{s.t.} \quad x^{T}x=1, \qquad y^{T}y=1, \quad x\in \mathbb {R}^{m}, y\in\mathbb{R}^{n}. \end{gathered} $$

The outer product \(\lambda x\circ y\circ x\circ y\), where

$$ (\lambda x\circ y\circ x\circ y)_{i_{1}i_{2}i_{3}i_{4}}=\lambda x_{i_{1}}y_{i_{2}}x_{i_{3}}y_{i_{4}},\quad \forall i_{1}, i_{3}\in[m], \forall i_{2}, i_{4}\in[n] $$

and λ is an M-eigenvalue with the maximal absolute value of \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{m\times n\times m\times n}\) with left M-eigenvector \(x\in\mathbb{R}^{m}\) and right M-eigenvector \(y\in\mathbb{R}^{n}\), is a partially symmetric best rank-one approximation of \(\mathcal {A}\) [1], which has wide applications in signal and image processing, wireless communication systems, and independent component analysis [11–14]. The M-spectral radius of \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in \mathbb{R}^{m\times n\times m\times n}\), denoted by

$$ \rho_{{\mathrm{M}}}(\mathcal{A})=\max\bigl\{ |\lambda|: \lambda \text{ is an M-eigenvalue of } \mathcal{A}\bigr\} , $$

has significant impacts on identifying nonsingular \(\mathscr {M}\)-tensors, which satisfy the strong ellipticity condition [10].

To our knowledge, there are few results about bounds for the M-spectral radius of a fourth-order partially symmetric tensor. In this paper, we present two bounds for the M-spectral radius and discuss their relation. A numerical example is also given to explain the proposed results.

2 Two bounds for the M-spectral radius

In this section, we give two bounds for the M-spectral radius of fourth-order partially symmetric tensors and discuss their relation.

Theorem 1

Let \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{m\times n\times m\times n}\) be a partially symmetric tensor. Then

$$ \rho_{{\mathrm{M}}}(\mathcal{A})\leq\sqrt{\max _{i\in[m]}\bigl\{ R_{i}(\mathcal{A})\bigr\} \cdot\max _{l\in[n]}\bigl\{ C_{l}(\mathcal{A})\bigr\} }, $$
(2)

where

$$ R_{i}(\mathcal{A}) =\sum_{i_{3}\in[m]}\sum _{i_{2}, i_{4}\in [n]}|a_{ii_{2}i_{3}i_{4}}|, \qquad C_{l}( \mathcal{A})=\sum_{i_{1}, i_{3}\in [m]}\sum _{i_{2}\in[n]}|a_{i_{1}i_{2}i_{3}l}|. $$

Proof

Suppose that λ is an M-eigenvalue of \(\mathcal{A}\) and that \(x\in\mathbb{R}^{m}\) and \(y\in\mathbb{R}^{n}\) are associated left M-eigenvector and right M-eigenvector. Then (1) holds. Let

$$ x_{p}=\max_{k\in[m]}\bigl\{ |x_{k}|\bigr\} ,\qquad y_{q}=\max_{k\in[n]}\bigl\{ |y_{k}|\bigr\} . $$

Since \(x^{T}x=1\) and \(y^{T}y=1\), we have

$$ 0< |x_{p}|\leq1,\qquad 0< |y_{q}|\leq1. $$
(3)

The pth equation of \(\mathcal{A}\cdot yxy=\lambda x\) is

$$ \lambda x_{p}=\sum_{i_{3}\in[m]}\sum _{i_{2},i_{4}\in [n]}a_{pi_{2}i_{3}i_{4}}y_{i_{2}}x_{i_{3}}y_{i_{4}}. $$
(4)

Taking the absolute values on both sides of (4) and using the triangle inequality give

$$ \begin{aligned}[b] |\lambda||x_{p}|&\leq\sum _{i_{3}\in[m]}\sum_{i_{2},i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||y_{i_{2}}||x_{i_{3}}||y_{i_{4}}| \\ &\leq\sum_{i_{3}\in[m]}\sum_{i_{2},i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||y_{q}| \\ &=R_{p}(\mathcal{A})|y_{q}|. \end{aligned} $$
(5)

Similarly, by the qth equation of \(\mathcal{A}xyx\cdot=\lambda y\) we have

$$ \begin{aligned}[b] |\lambda||y_{q}|&\leq\sum _{i_{1},i_{3}\in[m]}\sum_{i_{2}\in [n]}|a_{i_{1}i_{2}i_{3}q}||x_{i_{1}}||y_{i_{2}}||x_{i_{3}}| \\ &\leq\sum_{i_{1},i_{3}\in[m]}\sum_{i_{2}\in [n]}|a_{i_{1}i_{2}i_{3}q}||x_{p}| \\ &=C_{q}(\mathcal{A})|x_{p}|. \end{aligned} $$
(6)

Multiplying (5) and (6) gives

$$ |\lambda|^{2}|x_{p}||y_{q}|\leq R_{p}(\mathcal{A})C_{q}(\mathcal {A})|x_{p}||y_{q}|, $$

which, together with (3), yields

$$ |\lambda|^{2}\leq R_{p}(\mathcal{A})C_{q}( \mathcal{A})\leq\max_{i\in [m],l\in[n]}\bigl\{ R_{i}( \mathcal{A})C_{l}(\mathcal{A})\bigr\} . $$
(7)

Since (7) holds for all M-eigenvalues of \(\mathcal{A}\), we have

$$\rho_{{\mathrm{M}}}(\mathcal{A})\leq\sqrt{\max_{ i\in[m], l\in[n]}\bigl\{ R_{i}(\mathcal{A})C_{l}(\mathcal{A})\bigr\} }=\sqrt{\max _{i\in[m]}\bigl\{ R_{i}(\mathcal{A})\bigr\} \cdot\max _{l\in[n]}\bigl\{ C_{l}(\mathcal{A})\bigr\} }, $$

and the conclusion follows. □

Theorem 2

Let \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{m\times n\times m\times n}\) be a partially symmetric tensor, and let α be any subset of \([m]\) and β be any subset of \([n]\). Then

$$ \rho_{{\mathrm{M}}}(\mathcal{A})\leq\min\{\mu_{1}, \mu_{2}\}, $$
(8)

where

$$\begin{gathered} \mu_{1}=\min_{\alpha\subseteq[m]} \biggl\{ \max _{p\in[m], q\in[n]} \biggl\{ \frac{1}{2} \bigl(R_{p}^{\alpha}( \mathcal{A})+\sqrt{R_{p}^{\alpha }(\mathcal{A})^{2}+4 \bigl(R_{p}(\mathcal{A})-R_{p}^{\alpha}(\mathcal {A}) \bigr)C_{q}(\mathcal{A})} \bigr) \biggr\} \biggr\} , \\ \mu_{2}=\min_{\beta\subseteq[n]} \biggl\{ \max_{p\in[m], q\in[n]} \biggl\{ \frac{1}{2} \bigl(C_{q}^{\beta}(\mathcal{A})+ \sqrt{C_{q}^{\beta }(\mathcal{A})^{2}+4 \bigl(C_{q}(\mathcal{A})-C_{q}^{\beta}(\mathcal {A}) \bigr)R_{p}(\mathcal{A})} \bigr) \biggr\} \biggr\} ,\end{gathered} $$

and

$$ R_{p}^{\alpha}(\mathcal{A})=\sum_{i_{3}\in\alpha} \sum_{i_{2},i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}|,\qquad C_{q}^{\beta}( \mathcal{A})=\sum_{i_{2}\in \beta}\sum _{i_{1},i_{3}\in[m]}|a_{i_{1}i_{2}i_{3}q}|. $$

Proof

Assume that λ is an M-eigenvalue of \(\mathcal{A}\) and that \(x\in\mathbb{R}^{m}\) and \(y\in\mathbb{R}^{n}\) are the corresponding left M-eigenvector and right M-eigenvector. Then (1) holds. Let

$$ |x_{p}|=\max_{k\in[m]}\bigl\{ |x_{k}|\bigr\} ,\qquad |y_{q}|=\max_{k\in[n]}\bigl\{ |y_{k}|\bigr\} . $$

Then (3) holds. The pth equation of \(\mathcal{A}\cdot yxy=\lambda x\) can be rewritten as

$$ \lambda x_{p}=\sum_{i_{3}\in\alpha} \sum_{i_{2}, i_{4}\in [n]}a_{pi_{2}i_{3}i_{4}}y_{i_{2}}x_{i_{3}}y_{i_{4}}+ \sum_{i_{3}\notin \alpha}\sum_{i_{2},i_{4}\in[n]}a_{pi_{2}i_{3}i_{4}}y_{i_{2}}x_{i_{3}}y_{i_{4}}. $$
(9)

By the technique for the inequality in Theorem 1, we obtain from (9) that

$$ \begin{aligned} |\lambda||x_{p}|&\leq\sum _{i_{3}\in\alpha}\sum_{i_{2}, i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||y_{i_{2}}||x_{p}||y_{i_{4}}|+ \sum_{i_{3}\notin\alpha}\sum_{i_{2},i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||y_{i_{2}}||x_{i_{3}}||y_{q}| \\ &\leq\sum_{i_{3}\in\alpha}\sum_{i_{2}, i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||x_{p}|+ \sum_{i_{3}\notin\alpha}\sum_{i_{2},i_{4}\in[n]}|a_{pi_{2}i_{3}i_{4}}||y_{q}| \\ &=R_{p}^{\alpha}(\mathcal{A})|x_{p}|+ \bigl(R_{p}(\mathcal{A})-R_{p}^{\alpha }(\mathcal{A}) \bigr)|y_{q}|, \end{aligned} $$

that is,

$$ \bigl(|\lambda|-R_{p}^{\alpha}(\mathcal{A}) \bigr)|x_{p}|\leq\bigl(R_{p}(\mathcal {A})-R_{p}^{\alpha}( \mathcal{A})\bigr)|y_{q}|. $$
(10)

In addition, by the qth equation of \(\mathcal{A}xyx\cdot=\lambda y\) we have

$$ |\lambda||y_{q}|\leq\sum_{i_{1},i_{3}\in[m]} \sum_{i_{2}\in [n]}|a_{i_{1}i_{2}i_{3}q}||x_{p}|=C_{q}( \mathcal{A})|x_{p}|. $$
(11)

Multiplying (10) with (11) and using (3) yield

$$ \bigl(|\lambda|-R_{p}^{\alpha}(\mathcal{A}) \bigr)|\lambda|\leq\bigl(R_{p}(\mathcal {A})-R_{p}^{\alpha}( \mathcal{A})\bigr)C_{q}(\mathcal{A}). $$
(12)

Then

$$ \begin{aligned}[b] |\lambda|&\leq\frac{1}{2} \bigl(R_{p}^{\alpha}(\mathcal{A})+\sqrt{R_{p}^{\alpha}( \mathcal{A})^{2}+4\bigl(R_{p}(\mathcal{A}) -R_{p}^{\alpha}(\mathcal{A})\bigr)C_{q}(\mathcal{A})} \bigr) \\ &\leq\max_{p\in[m],q\in[n]} \biggl\{ \frac{1}{2} \bigl(R_{p}^{\alpha }(\mathcal{A})+\sqrt{R_{p}^{\alpha}( \mathcal{A})^{2}+ 4\bigl(R_{p}(\mathcal{A})-R_{p}^{\alpha}( \mathcal{A})\bigr)C_{q}(\mathcal {A})} \bigr) \biggr\} . \end{aligned} $$
(13)

Note that (13) holds for all M-eigenvalues of \(\mathcal{A}\) and any \(\alpha\subseteq[m]\). Hence

$$ \rho_{{\mathrm{M}}}(\mathcal{A})\leq\mu_{1}. $$
(14)

On the other hand, for the qth equation of \(\mathcal{A}xyx\cdot =\lambda y\), we have

$$ \lambda y_{q}=\sum_{i_{2}\in\beta} \sum_{i_{1},i_{3}\in [m]}a_{i_{1}i_{2}i_{3}q}x_{i_{1}}y_{i_{2}}x_{i_{3}}+ \sum_{i_{2}\notin\beta}\sum_{i_{1},i_{3}\in [m]}a_{i_{1}i_{2}i_{3}q}x_{i_{1}}y_{i_{2}}x_{i_{3}}. $$
(15)

Then

$$ \begin{aligned} |\lambda||y_{q}|&\leq\sum _{i_{2}\in\beta}\sum_{i_{1},i_{3}\in [m]}|a_{i_{1}i_{2}i_{3}q}||y_{q}|+ \sum_{i_{2}\notin\beta}\sum_{i_{1},i_{3}\in[m]}|a_{i_{1}i_{2}i_{3}q}| |x_{p}| \\ &=C_{q}^{\beta}(\mathcal{A})|y_{q}|+ \bigl(C_{q}(\mathcal{A})-C_{q}^{\beta }(\mathcal{A}) \bigr)|x_{p}|, \end{aligned} $$

that is,

$$ \bigl(|\lambda|-C_{q}^{\beta}(\mathcal{A}) \bigr)|y_{q}|\leq\bigl(C_{q}(\mathcal {A})-C_{q}^{\beta}( \mathcal{A})\bigr)|x_{p}|. $$
(16)

By the pth equation of \(\mathcal{A}\cdot yxy=\lambda x\) we have

$$ |\lambda||x_{p}|\leq\sum_{i_{3}\in[m]} \sum_{i_{2},i_{4}\in [n]}|a_{pi_{2}i_{3}i_{4}}||y_{q}|=R_{p}( \mathcal{A})|y_{q}|. $$
(17)

Multiplying (16) with (17) and using (3), we derive

$$ \bigl(|\lambda|-C_{q}^{\beta}(\mathcal{A}) \bigr)|\lambda|\leq\bigl(C_{q}(\mathcal {A})-C_{q}^{\beta}( \mathcal{A})\bigr)R_{p}(\mathcal{A}). $$
(18)

Hence

$$ \begin{aligned}[b] |\lambda|&\leq\frac{1}{2} \bigl(C_{q}^{\beta}(\mathcal{A})+\sqrt{C_{q}^{\beta}( \mathcal{A})^{2}+4\bigl(C_{q}(\mathcal{A})-C_{q}^{\beta }( \mathcal{A})\bigr)R_{p}(\mathcal{A})} \bigr) \\ &\leq\max_{p\in[m], q\in[n]} \biggl\{ \frac{1}{2} \bigl(C_{q}^{\beta }(\mathcal{A})+\sqrt{C_{q}^{\beta}( \mathcal{A})^{2}+4\bigl(C_{q}(\mathcal {A})-C_{q}^{\beta}( \mathcal{A})\bigr)R_{p}(\mathcal{A})} \bigr) \biggr\} . \end{aligned} $$
(19)

Since (19) holds for all M-eigenvalues of \(\mathcal{A}\) and any \(\beta\subseteq[n]\), we have

$$ \rho_{{\mathrm{M}}}(\mathcal{A})\leq\mu_{2}. $$
(20)

From (14) and (20) we have

$$ \rho_{{\mathrm{M}}}(\mathcal{A})\leq\min\{\mu_{1}, \mu_{2} \}. $$

The proof is completed. □

Remark 1

Since

$$ \begin{gathered} \max_{p\in[m], q\in[n]} \biggl\{ \frac{1}{2} \bigl(R_{p}^{\alpha}(\mathcal {A})+ \sqrt{R_{p}^{\alpha}(\mathcal{A})^{2}+4 \bigl(R_{p}(\mathcal {A})-R_{p}^{\alpha}(\mathcal{A}) \bigr)C_{q}(\mathcal{A})} \bigr) \biggr\} \\ \quad=\max_{p\in[m], q\in[n]} \biggl\{ \frac{1}{2} \bigl(C_{q}^{\beta }(\mathcal{A})+\sqrt{C_{q}^{\beta}( \mathcal{A})^{2}+4\bigl(C_{q}(\mathcal {A})-C_{q}^{\beta}( \mathcal{A})\bigr)R_{p}(\mathcal{A})} \bigr) \biggr\} \\ \quad=\sqrt{\max_{p\in[m]}\bigl\{ R_{p}(\mathcal{A}) \bigr\} \cdot\max_{q\in [n]}\bigl\{ C_{q}(\mathcal{A})\bigr\} } \end{gathered} $$

when \(\alpha=\varnothing\) and \(\beta=\varnothing\), we have

$$ \min\{\mu_{1}, \mu_{2}\}\leq\sqrt{\max _{p\in[m]}\bigl\{ R_{p}(\mathcal {A})\bigr\} \cdot\max _{q\in[n]}\bigl\{ C_{q}(\mathcal{A})\bigr\} }. $$

Therefore, the bound in (8) is tighter than the bound in (2) for the M-spectral radius \(\rho_{{\mathrm{M}}}(\mathcal{A})\) of a given tensor \(\mathcal{A}\).

Remark 2

Although the bound in (8) is tighter than the bound in (2), it is easier to compute the bound in (2) for the M-spectral radius of a given tensor.

Next, we use a numerical example to show the effectiveness of the bounds in Theorems 1 and 2.

Example 1

Consider the partially symmetric tensor \(\mathcal {A}_{1}=(a_{i_{1}i_{2}i_{3}i_{4}})\in\mathbb{R}^{3\times3\times 3\times3}\) with

$$ \begin{gathered} a_{1111}=1.1112,\qquad a_{1311}=6.1096,\qquad a_{3111}=0.3032,\qquad a_{2121}=1.4125, \\ a_{3131}=1,\qquad a_{1212}=0.0788,\qquad a_{2222}=1,\qquad a_{3222}=0.6032, \\ a_{3232}=0.3657,\qquad a_{1313}=2,\qquad a_{2323}=0.6226,\qquad a_{3333}=0.3, \end{gathered} $$

and the remaining zero elements. By Theorem 1 we have

$$ \rho_{{\mathrm{M}}}(\mathcal{A}_{1})\leq12.6843. $$

By Theorem 2 we have

$$ \rho_{{\mathrm{M}}}(\mathcal{A}_{1})\leq10.2397. $$

In fact, \(\rho_{{\mathrm{M}}}(\mathcal{A}_{1})\approx7.6841\).

3 Conclusions

In this paper, we have presented two bounds for the M-spectral radius of a fourth-order partially symmetric tensor and have indicated their relation. To show the effectiveness of the proposed results, a numerical example is also given.