Estimations for spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices
© Wang et al.; licensee Springer. 2014
Received: 9 March 2014
Accepted: 18 November 2014
Published: 1 December 2014
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.
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 ). Some spectral properties of these two classes of matrices are discussed in the paper.
Though the result is earlier than the Geršgorin theorem (see ), 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.
A Brauer (see ) gave the Brauer estimation for the spectral radius of nonnegative matrices by the Cassini oval region, and he improved Perron-Frobenius’ result.
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 , we can give the Brauer estimation for the M-matrix min-eigenvalues.
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 ) 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 ).
Definition 2.1 
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 
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 , .
- (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 .
(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.
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 .
- (i)Since , . From (1), we easily see
- (ii)Let such that . From (1), we easily see that
Combining (i) with (ii), we have .
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.
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.
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.
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.
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 .
Proof Let , , be the spectral radius of B and . Obviously . It follows from Theorem 3.1 that , and then .
Similarly, it follows that . So . □
Analogously, we have the following results.
If we take , we have the following corollary.
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.
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 .
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 .
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.
We wish to thank the anonymous referees for their thorough reading and constructive comments.
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