Journal of Applied Mathematics

Volume 2013 (2013), Article ID 985654, 5 pages

http://dx.doi.org/10.1155/2013/985654

Research Article

On Some Properties for the Sequence of Brualdi-Li Matrices

Xiaogen Chen

School of Information Science and Technology, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China

Received 16 August 2013; Accepted 5 December 2013

Academic Editor: Han H. Choi

Copyright © 2013 Xiaogen Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let denote the Brualdi-Li matrix, and let denote the Perron value of the Brualdi-Li matrix of order . We prove that is monotonically decreasing for all and , where

1. Introduction

A tournament matrix of order is a matrix satisfying the equation , where is the all ones matrix, is the identity matrix, and is the transpose of . If is odd, then the maximum possible Perron value of a tournament matrix of order is , attained when the row sums of are all equal [1]. However, if is even, it is difficult to attain. Brualdi and Li [2] conjectured that the matrix

where is strictly upper triangular tournament matrix (all of its entries above the main diagonal are equal to one), that is,

is the tournament matrix of order which maximizes the Perron value. This conjecture has recently been confirmed in [3].

The matrix has been dubbed by the Brualdi-Li matrix [4]. Let denote the Perron value of the Brualdi-Li matrix of order .

It is well known that .

Friedland [4] has paid particular attention to the behavior of the Perron value of . Friedland and Katz [5] posed the problem of determining the behavior of the sequence . Kirkland [6] conjectured that is monotonically decreasing for all . We investigate this conjecture and obtain the following results.

Theorem 1. Let be an integer, and let be the Perron value of the Brualdi-Li matrix . Then is monotonically decreasing for all .

By Theorem 1 and Lemma 4, one can easily prove the following corollaries.

Corollary 2. Let be an integer, and let be the Perron value of the Brualdi-Li matrix . Then

where .

In [6], Kirkland proved that

where .

We make the following conjecture and remark that it is supported by computations for .

Conjecture 3. is monotonically decreasing for all .

We give some lemmas in Section 2 and prove Theorem 1 in Section 3.

2. Some Lemmas

2.1. Intermediate Value Theorem

If is a real-valued continuous function on the closed interval , and is a number between and , then there is such that . where is open interval.

For two variables, the Implicit Function Theorem is as follows.

2.2. Implicit Function Theorem

If a real-valued function is defined on an open disk containing point , satisfying

(i),(ii),(iii) and are continuous on the open disk,

then the equation defines as a function of near the point and the derivative of this function is given by

where and indicate the partial derivatives of with respect to and .

Lemma 1 (see [6]). Let be an integer, and let be the Perron value of the Brualdi-Li matrix . Then

Let and , , and are real numbers}, then the partial derivatives of are

Evidently, and are derivable on the open disk , and  , . By the Implicit Function Theorem, the equation defines as a function in terms of and the derivative of this function is given by

Lemma 2. Let be a real number, and let be a function in terms of satisfying the equation . Then

where , .

Proof. For fixed real number , is continuous real function in terms of .

Let and . It is easy to show that the real functions and are monotonically decreasing for real number . Thus

Notice that

where . We have

Hence . By the Intermediate Value Theorem, there is such that ; that is,

We are done.

Lemma 3. Let be a real number, and let be a function in terms of satisfying the equation , and let , . Then

Proof. For fixed real number , is continuous real function in terms of . Let and ; then

By Lemma 2, we have

Hence

that is,

We have the desired result.

Lemma 4 (see [6]). There is an such that the sequence decreases monotonically to its limit for .

3. Proof of Theorem 1

In Section 2, and  , , and is the function in terms of satisfying the equation . Clearly, and are the functions in terms of .

Let be real number. Since , we have

Further

Thus

We have

Hence

Let and

then

Let and , then

Obviously,

By Lemma 2 and Taylor’s Theorem, we have

Due to , then

By Lemmas 2 and 3,

By Lemma 3 and (25)–(30), we have for . Hence , where is a real number.

To sum up, we obtain the following result.

Let be a real number, and let be a function of satisfying the equation , ; then is monotonically decreasing function for all .

Note that is the Perron value of the Brualdi-Li matrix , by Lemma 1, satisfy the equation

where is an integer. It is easy to see that is monotonically decreasing for all .

Since

evidently, it follows that is monotonically decreasing for all integers . We complete the proof of Kirkland’s conjecture.

Acknowledgment

This work is supported by the Natural Science Foundation of Guangdong Province, China (no. S2012010009759).

References

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R. A. Brualdi and Q. Li, “Problem 31,” Discrete Mathematics, vol. 43, pp. 329–330, 1983.

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S. Friedland and M. Katz, “On the maximal spectral radius of even tournament matrices and the spectrum of almost skew-symmetric compact operators,” Linear Algebra and Its Applications, vol. 208, pp. 455–469, 1994.

S. Kirkland, “A note on the sequence of Brualdi-Li matrices,” Linear Algebra and Its Applications, vol. 248, pp. 233–240, 1996.