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Upper bounds for the spectral radius of matrices having the Perron–Frobenius property
Journal of Inequalities and Applications volume 2024, Article number: 68 (2024)
Abstract
A new upper bound for the spectral radius of matrices having the Perron–Frobenius property is given by considering the position of positive entries. Some examples involving the largest zero of polynomials and the spectral radius of the iterative matrix for the Perron–Frobenius splitting are given to show the superiority of the theoretical result.
1 Introduction
Matrices having the Perron–Frobenius property [7] arise in many different fields of science and engineering, such as steady state behavior of Markov chains, population growth models, and Web search engines [2, 9, 11, 12].
Definition 1
[7] If \(A\in \mathbb{R}^{n\times n}\), then A possesses the Perron–Frobenius (P–F) property if the spectral radius
is a positive eigenvalue of A and \(\rho (A) \) possesses a corresponding nonnegative eigenvector, where \(\sigma (A)\) is the spectrum of matrix A.
The well-known classes of nonnegative matrices, eventually positive matrices, and eventually nonnegative matrices [1, 3] are all included in matrices having the P–F property. There are various problems on matrices having the P–F property, for details, see [8, 10, 13, 14]. One of such problems is to bound its dominant eigenvalue \(\rho (A)\), and the first result stated as below is due to Noutsos in 2006 [7].
Theorem 1
[7, Theorem 2.5] Let \(A=(a_{ij})\in \mathbb{R}^{n\times n}\) be a matrix having the P–F property. Then
where \(N=\{1,2,\ldots ,n\}\) and \(R_{i}(A^{\top})=\sum_{j\in N} a_{ij}\).
The upper bound (1) due to Noutsos for matrices having the P–F property was improved by He, Liu, and Lv in 2023 by using the positive part of \(R_{i}(A)\) [4].
Theorem 2
[4, Theorems 2, 3 and 5] Let \(A=(a_{ij})\in \mathbb{R}^{n\times n}\) be a matrix having the P–F property. Then
and
Furthermore, \(Bnd_{HLL_{2}}(A)\leq Bnd_{HLL_{1}}(A)\), where \(r_{i}^{+}(A):=\sum_{j\in N,~j\neq i \atop a_{ij}>0} a_{ij}\).
Besides bounds (2) and (3), He et al. also provided an S-type upper bound for the spectral radius in [4], but this bound needs more computations. In this paper, we present a new upper bound for the spectral radius for matrices having the P–F property, and show that the new bound is sharper than bounds (2) and (3). Some numerical examples are also given to show the superiority of the new bound.
2 Main results
In this section we give a new upper bound for the spectral radius of matrices having the P–F property.
Theorem 3
Let \(A=(a_{ij})\in \mathbb{R}^{n\times n}\) be a matrix having the Perron–Frobenius property, and \(r_{i}^{+}(A) >0 \) for any \(i\in N\). Then
where \(Bnd(A):= \frac{1}{2}\max_{i,j\in N, j\neq i \atop a_{ij} > 0} (a_{ii}+a_{jj}+ ((a_{ii}-a_{jj})^{2}+4r_{i}^{+}(A)r_{j}^{+}(A) )^{\frac{1}{2}} )\).
Proof
Let \(\mathbf{x}=(x_{1},x_{2},\ldots ,x_{n})^{\top}\) be an entrywise nonnegative nonzero eigenvector of matrix A corresponding to \(\rho (A)\), that is,
Let
From (5), we have that, for any \(i\in N\),
and
Case I. If \(x_{i_{0}} x_{j_{0}} > 0\), then \(x_{i_{0}} >0\) and \(x_{j_{0}} > 0\). By (7), we have
and
Case I-a. If \(\rho (A)-a_{i_{0}i_{0}} > 0\) and \(\rho (A)-a_{j_{0}j_{0}} > 0\), then multiplying (8) by (9) gives
and hence
Solving \(\rho (A)\) in (10) gives
i.e., inequality (4) holds.
Case I-b. If \(\rho (A)-a_{i_{0}i_{0}}\leq 0\) or \(\rho (A)-a_{j_{0}j_{0}}\leq 0\), then \(\rho (A)\leq \max \{a_{i_{0}i_{0}}, a_{j_{0}j_{0}}\}\). Without loss of generality, suppose that \(a_{i_{0}i_{0}} \geq a_{j_{0}j_{0}}\), then
i.e., inequality (4) holds.
Case I-c. If \(\rho (A)-a_{i_{0}i_{0}}\leq 0\) and \(\rho (A)-a_{j_{0}j_{0}}\leq 0\), then \(\rho (A)\leq \min \{a_{i_{0}i_{0}}, a_{j_{0}j_{0}}\}\), and thus \(\rho (A)\leq \min \{a_{i_{0}i_{0}}, a_{j_{0}j_{0}}\} \leq \max \{a_{i_{0}i_{0}}, a_{j_{0}j_{0}}\}\). From Case I-b, inequality (4) also holds.
Case II. If \(x_{i_{0}} x_{j_{0}} = 0\), then from \(\mathbf{x}\neq 0 \), there exists one index \(k\in N\) such that \(x_{k}\neq 0\). For this index k, by the assumption that \(r_{i}^{+}(A) >0 \) for any \(i\in N\), we have \(r_{k}^{+}(A) >0\), and hence there is an index \(k_{0}\in N\) and \(k_{0}\neq k\) such that \(a_{kk_{0}} > 0\). Furthermore, by (7) it follows that
and hence \(\rho (A)\leq a_{kk}\). Similarly to Case I-b and Case I-c, we have
i.e., inequality (4) also holds. The conclusion follows from Case I and Case II. □
Remark here that the difference of bounds (3) and (4) is the restriction under the max, from which it holds obviously that \(Bnd(A) \leq Bnd_{HLL_{2}} \leq Bnd_{HLL_{1}}\), and the bound \(Bnd(A)\) need less computations than \(Bnd_{HLL_{2}}\) in general.
For a nonnegative matrix A, it holds that \(r_{i}(A)=r_{i}^{+}(A)\) for any \(i\in N\). Hence, we can obtain the following bound for the spectral radius for nonnegative matrices because a nonnegative matrix is a matrix having the P–F property.
Corollary 1
Let \(A=(a_{ij})\in \mathbb{R}^{n\times n}\) be a nonnegative matrix, and \(r_{i}(A) >0 \) for any \(i\in N\). Then
where \(Bnd_{N}(A):= \frac{1}{2}\max_{i,j\in N, j\neq i \atop a_{ij} \neq 0} (a_{ii}+a_{jj}+ ((a_{ii}-a_{jj})^{2}+4r_{i}(A)r_{j}(A) )^{\frac{1}{2}} )\).
Remark here that the upper bound (11) for the spectral radius of nonnegative matrices is exactly the bound provided by Kolotilina [5].
Example 1
Consider the matrix
with the spectral radius \(\rho (A)=3.4292\) and the corresponding eigenvector \(\mathbf{x}=(0.3107, 0.3707, 0.6992, 0.3082,0.4269)^{\top}\). Bound (1) in Theorem 1, bounds (2) and (3) in Theorem 2, and our bound (4) in Theorem 3 are listed in Table 1. From Table 1 it can be seen that our bound (4) is sharper than bounds (2) and (3), and sharper than bound (1) in some cases.
Example 2
Consider the matrix
and the Perron–Frobenius splitting [7, 8] of A with \(A=M-N\), where
and the iterative matrix
has the P–F property with the spectral radius \(\rho (M^{-1}N)=0.5392\) and the corresponding eigenvector \(\mathbf{x}=(0.4460, 0.4497, 0.7663, 0.1077)^{\top}\). Bounds (1), (2), (3), and (4) for \(\rho (M^{-1}N)\) are given in Table 2. From Table 2 it can be seen that bounds (1), (2), and (3) are larger than 1. However our bound (4) is less than 1, which implies that we can conclude that the Perron–Frobenius splitting for this case is convergent by our bound.
Example 3
Consider the matrix
which is the companion matrix of the polynomial
Matrix A has the P–F property with the dominant eigenvalue \(\rho (A)=1.1453\) and the corresponding eigenvector \(\mathbf{x}=(0.3872, 0.5141, 0.4137, 0.5020, 0.4032)^{\top}\), where \(\rho (A)=1.1453\) is also the largest zero of the polynomial \(p(z)\). Besides bounds (1), (2), (3), and (4), we give the upper bound proposed by Melman (see Theorem 3.1 of [6]) for zeros of the polynomial, see Table 3. From Table 3 it can be seen that our bound (4) is sharper than bounds (2) and (3), and sharper than bound (1) and Melman’s bound in Theorem 3.1 of [6] in some cases.
3 Conclusions
In this paper we propose a new upper bound for the spectral radius by considering the position of the positive entries for a given matrix having the Perron–Frobenius property. We conjecture that by this technique the S-type upper bound for the spectral radius in [4] can be improved further.
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References
Berman, A., Catral, M., DeAlba, L., et al.: Sign patterns that allow eventual positivity. Electron. J. Linear Algebra 19, 108–120 (2009)
Diao, W.L., Ma, C.Q.: Sign-consensus of linear multiagent systems under a state observer protocol. Complexity 2019, 3010465 (2019)
Elhashash, A., Szyld, D.B.: Two characterizations of matrices with the Perron–Frobenius property. Numer. Linear Algebra Appl. 16(11–12), 863–869 (2009)
He, J., Liu, Y., Lv, W.: New upper bounds for the dominant eigenvalue of a matrix with Perron–Frobenius property. J. Inequal. Appl. 2023, 13 (2023)
Kolotilina, L.Y.: Bounds and inequalities for the Perron root of a nonnegative matrix. Zap. Nauč. Semin. POMI 284, 77–122 (2002)
Melman, A.: Upper and lower bounds for the Perron root of a nonnegative matrix. Linear Multilinear Algebra 61(2), 171–181 (2013)
Noutsos, D.: On Perron–Frobenius property of matrices having some negative entries. Linear Algebra Appl. 412(2–3), 132–153 (2006)
Noutsos, D.: On Stein–Rosenberg type theorems for nonnegative and Perron–Frobenius splittings. Linear Algebra Appl. 429(8–9), 1983–1996 (2008)
Noutsos, D., Tsatsomeros, M.J.: Reachability and holdability of nonnegative states. SIAM J. Matrix Anal. Appl. 30(2), 700–712 (2008)
Olesky, D., Tsatsomeros, M., Van den Driessche, P.: \(M_{\vee}\)-Matrices: a generalization of M-matrices based on eventually nonnegative matrices. Electron. J. Linear Algebra 18, 339–351 (2009)
Pillai, S.U., Suel, T., Cha, S.: The Perron–Frobenius theorem: some of its applications. IEEE Signal Process. Mag. 22(2), 62–75 (2005)
Sootla, A.: Properties of eventually positive linear input–output systems. IET Control Theory Appl. 13(7), 891–897 (2019)
Tarazaga, P.: On the structure of the set of symmetric matrices with the Perron–Frobenius property. Linear Algebra Appl. 549, 219–232 (2018)
Tarazaga, P., Raydan, M., Hurman, A.: Perron–Frobenius theorem for matrices with some negative entries. Linear Algebra Appl. 328(1–3), 57–68 (2001)
Funding
This work was supported by Yunnan Fundamental Research Projects (Grant NO. 202401AT070479), and Yunnan Provincial Xingdian Talent Support Program.
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K.M. Li (first author): conceptualization, methodology, writing-original draft; C.Q. Li (Second author, corresponding author): conceptualization, writing-original draft, writing review and editing.
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Li, K., Li, C. Upper bounds for the spectral radius of matrices having the Perron–Frobenius property. J Inequal Appl 2024, 68 (2024). https://doi.org/10.1186/s13660-024-03144-y
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DOI: https://doi.org/10.1186/s13660-024-03144-y