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Estimations for spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices
Journal of Inequalities and Applications volume 2014, Article number: 477 (2014)
Abstract
In this paper, some estimations for the spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices are given by matrix directed graphs and their k-path covering. The existent results on the upper and lower bounds of the spectral radius of nonnegative matrices are improved.
MSC:15A18, 65F15.
1 Introduction
The following notations are used throughout this paper. Let be an matrix with real entries. Denote , , , . denotes the prime sub-matrices of A where row-column subscripts are all in , is the spectral radius of A, and is the smallest eigenvalue according to the module of A.
For , if , , A is called a nonnegative matrix. We denote it . If , , , A is called a nonsingular M-matrix. We denote it . Nonnegative matrices and M-matrices are two important matrix classes, which are applied in many fields such as computational mathematics, probability theory, and mathematical economics (see [1]). Some spectral properties of these two classes of matrices are discussed in the paper.
With regard to estimations for the nonnegative matrix spectral radius, the earliest result is given by Perron-Frobenius (see [2]), that is,
Though the result is earlier than the Geršgorin theorem (see [3]), it can be seen as the estimation of by using the right end-point of the Geršgorin disc. Therefore it is still called a Geršgorin estimation.
Denote
A Brauer (see [4]) gave the Brauer estimation for the spectral radius of nonnegative matrices by the Cassini oval region, and he improved Perron-Frobenius’ result.
Suppose is inducible, then
A main equivalent representation of M-matrices indicates that the real parts of M-matrix eigenvalues are all positive (see [1]). Tong (see [5]) improved the result and concluded that the min-eigenvalue by the module of an M-matrix is a positive number. Zhang (see [6]) proved that the min-eigenvalue of an M-matrix by its real part is also its min-eigenvalue by the module and offered the estimation formula
For the same reason, we also call it a Geršgorin type estimation. Let A be a nonsingular M-matrix and denote . Then , , thus , and the above estimation results as regards the M-matrix are natural. Therefore, according to [4], we can give the Brauer estimation for the M-matrix min-eigenvalues.
Denote
Let A be a nonsingular and irreducible M-matrix. Then
In this paper, we present the Brualdi type estimation for the nonnegative matrix spectral radius and the minimal eigenvalue of M-matrix by the deduction method with directed graph (see [7]) and the concept of the k-path covering of the directed graph. Moreover, we give the improved Brauer type estimations, which improve the relevant results of [4–6, 8–10].
2 Directed graph and its k-path covering
Let be the directed graph of . denotes its nodal set, and denotes its directed edge set. The directed edge sequence γ: is called the simple loop of where and all are different from each other. In short, we denote γ: . Let be the length of the simple loop , then is the set of all the simple loops of matrix A.
In this paper, directed graphs and simple loops of are all regarded as its sub-graphs, and means i’s successor set in .
We will introduce the concept of k-path coverage of matrix directed graphs (see [11]).
Definition 2.1 [11]
Let γ: , k be an integer, , and be the least common multiple of t and s. We call the set made up of directed paths in γ, respectively, starting with , and including directed edges as the k-path coverage of the simple loop γ ( as base point), denoted by .
Definition 2.2 [11]
For every in , is called a k-path coverage in .
Obviously, if the chosen base-points are different, usually there are different k-path coverings . When , , the k-path coverage on is γ itself. If , , then only covers every nodal point on γ once and there are different k-path coverings on γ. If , then .
Besides, when , let γ: , η be the maximal common divisor of 2 and s, and . The set is called an odd 1-path covering of γ (corresponding to the contemporary notation); and the set is called an even 1-path coverage of γ (corresponding to the contemporary notation). Likewise, a definitive 1-path coverage of γ is denoted as . When s is a positive odd number, the even 1-path and the odd one of γ are the same. Namely there is only a 1-path coverage including all s directed edges in γ. When s is a positive even number, there are two odd and even 1-path coverings, respectively, including its pieces of directed edges in γ.
A relation ‘≺’ on the nonempty set ν is called pre-order, if ‘≺’ satisfies reflexivity and transitivity, namely , ; and implies , .
Let ‘≺’ be a pre-order on the nodal set of , . If for all , , then:
-
(1)
If for all , , then there exists a simple loop γ: such that
(2.1) -
(2)
If is a k-path covering of the above simple loop, then there exists a directed path ρ: in γ, such that
(2.2)(2.3)
Proof (1) See [7].
(2) Without loss of generality, we suppose that is the base-point of , , , and : for . Let be the largest element of the nodal set in the sense of pre-order ‘≺’. In γ, take a directed path ρ: starting with including directed edges, then . According to (2.1), then (2.2) holds. Because is the largest element of and , (2.3) holds. □
The discussion in this paper needs some basic results as regards a reducible matrix determined by a polynomial.
If A is a reducible matrix with , then there exists a permutation matrix P such that
where is principal sub-matrix of A and . They are either irreducible or zero matrix with order 1, . The right-side matrix of (2.4) is called the reduced polynomial of A. If the order of is not considered, (2.4) has nothing to do with the choice of P. So (2.4) is a unique definite partition on a set corresponding to the subscripted set of . When A is an irreducible matrix and a zero matrix with order 1, for unity, denote , where and . Denote
Obviously .
Definition 2.3 Let . If , then A is a weakly irreducible matrix, which is denoted by . For a general matrix, if , we call the weakly irreducible nucleus of A, which is denoted by . Denote if .
3 The spectral radius of a nonnegative matrix
In this section, we suppose that .
Lemma 3.1 Let , , . Define a function . Then is strongly monotone increasing when .
Theorem 3.1 Let . , denotes the real roots of which are larger than . Denote
then
Proof (1) A is irreducible. By the Perron-Frobenius theorem, we know there is , such that . Define the pre-order ‘≺’: on the nodal set of , if and only if . From Lemma 2.1(1), there exists : such that , , . Hence, from
we obtain
Then
i.e.
Similarly, if we define a pre-order ‘≺’: on the nodal set of , if and only if . From Lemma 2.1, there exists : such that , , . Similar to the above, we obtain
Besides, notice and . From (3.1) and (3.2), it is deduced that
i.e. .
(2) A is weakly irreducible. It can be supposed that A has got its polynomial (2.4), where is irreducible whose order is not less than 2. We prove it with the following two steps.
-
(i)
Since , . From (1), we easily see
-
(ii)
Let such that . From (1), we easily see that
Combining (i) with (ii), we have .
(3) A is weakly irreducible. Noticing , from (2), we see
Since
i.e. . □
Remark 3.1 Because and have relations to the directed graphs, when A is reducible, and traditional continuity deduction has no effect, we use (2.4) to prove it. Besides, in Theorem 3.1, must be defined by , but it cannot be defined by directly. See Example 5.3.
Theorem 3.2 Let and be the k-path covering of . , always denote its nodal set as ρ: . denotes the real roots of the equation , . Denote
then
Proof Let A be an irreducible and nonnegative matrix. By the Perron-Frobenius theorem, there exists such that . Define a pre-order ‘≺’: on the nodal set of , if and only if . It follows from Lemma 2.1 that there exists a directed path : in , such that , , ; ; , . Because , we get
Multiply all the inequalities in (3.3):
Furthermore,
Then
Likewise, a pre-order ‘≺’: is defined on the nodal set of if and only if . It follows from Lemma 2.1 that there exists a directed path : in such that , , ; , and , . Therefore from , we obtain
Thus, if is irreducible, with (3.4) and (3.5), the theorem is proved. If is a weakly irreducible or non-weakly irreducible matrix, similar to the proof of (2), (3) in Theorem 3.1, it is the same with the theorem here. □
In Theorem 3.2, if , it is Theorem 3.1. Therefore Theorem 3.1 can be viewed as a special case of Theorem 3.2.
Theorem 3.3 Let , and be the k-path covering of . , its nodal set is always denoted as ρ: . The other notations are the same as those in Theorem 3.1 and Theorem 3.2. Then
Proof First we prove . It is necessary to prove that for all there exists such that . Otherwise there exists γ: , , such that , . Note that is the real root of the equation , larger than . From Lemma 3.1, we see that
Multiply all the inequalities in (3.6),
i.e.
Furthermore
Then from Lemma 3.1, we know that (3.7) implies , which leads to a contradiction.
Similarly, can be proved. With the above and Theorem 3.1, the theorem is proved. □
If , the following result, which is more convenient to use, can be obtained from Theorem 3.2.
Corollary 3.1 Let , and be the 1-path covering of . Denote
Then
Obviously, generally speaking, different k-path coverings can be taken for the directed graph of . We denote the congregation set of these different k-path coverings of as , then we have the following theorem.
Theorem 3.4 Let , for a given k-path covering of , with , , the same as in Theorem 3.2. Then
Remark 3.2 Theorem 3.1 can be viewed as the estimation for by means of the right end-point of the Brualdi region of an eigenvalue distribution, so it is called a Brualdi estimation. Corollary 3.1 is the improved Brauer estimation. Because we only need to calculate the corresponding of the edge in the simple loop, especially when the length of the loop is even, only the corresponding of half of the edges needs to be calculated. Thus the calculation decreases greatly and meanwhile the accuracy improves. If Theorem 3.4 is used, a general estimation is more accurate. Theorem 3.1 and Corollary 3.1 are both superior to the results of Perron-Frobenius and Brauer. See Example 5.1 and Example 5.2.
4 The smallest eigenvalue of M-matrix
In this section, we define .
Theorem 4.1 Let be a nonsingular M-matrix. , is the real root of the equation , and . Denote
Then
Proof Let , , be the spectral radius of B and . Obviously . It follows from Theorem 3.1 that , and then .
Define , . Then it follows from the definitions of and that . We obtain
Similarly, it follows that . So . □
Analogously, we have the following results.
Theorem 4.2 Let be a nonsingular M-matrix, and be the k-path covering of . , always denote its nodal set as ρ: . Denoted by the real root of the equation , which is less than , and denote
Then
and
If we take , we have the following corollary.
Corollary 4.1 Let be a nonsingular M-matrix, is the 1-path covering of . Denote
Then
Remark 4.1 Theorem 4.1 and Corollary 4.1 are, respectively, the Brualdi type estimation and the Brauer type estimation for the least eigenvalue of the M-matrix. These results are more accurate. See the matrix B in Example 5.1 and Example 5.2.
5 Examples
Example 5.1 Consider the nonnegative matrix
By calculating, . It follows from a Geršgorin type estimation that . Because , , , , and , it follows from a Brauer type estimation that . Take , and it follows from Corollary 3.1 that .
Consider a nonsingular M-matrix . It only follows from Geršgorin type and Brauer type estimations that . Take , and it follows from Corollary 4.1 that . But in fact .
Example 5.2 Consider the nonnegative matrix
By calculating, . It follows from a Geršgorin type estimation that . Because , , , , , and , it follows from a Brauer type estimation that . Take , and it follows from Corollary 3.1 that . Because , and it follows from Theorem 3.1 that the estimation is already accurate.
Consider the nonsingular M-matrix . It follows from a Geršgorin type estimation that and it follows from a Brauer type estimation that . Take , and it follows from Corollary 4.1 that . It follows from Theorem 4.1 that we obtain the accurate result .
Example 5.3 When A is weakly irreducible, , . Consider the following non-weakly irreducible and nonnegative matrix:
Obviously, , , and . In Theorem 3.1, if is defined as the real root of , which is larger than , then , and, moreover, . According to Theorem 3.1, obviously it is false. Likewise, if , , in Corollary 3.1, Theorem 4.1 and Corollary 4.1 are directly defined by ; mistakes also happen, which will not be discussed in detail.
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We wish to thank the anonymous referees for their thorough reading and constructive comments.
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Wang, T., Lv, H. & Sang, H. Estimations for spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices. J Inequal Appl 2014, 477 (2014). https://doi.org/10.1186/1029-242X-2014-477
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DOI: https://doi.org/10.1186/1029-242X-2014-477