- Research
- Open access
- Published:
Matrix spectral norm Wielandt inequalities with statistical applications
Journal of Inequalities and Applications volume 2014, Article number: 110 (2014)
Abstract
In this article, we construct a new matrix spectral norm Wielandt inequality. Then we apply it to give the upper bound of a new measure of association. Finally, a new alterative based on the spectral norm for the relative gain of the covariance adjusted estimator of parameters vector is given.
1 Introduction
Suppose that A is an positive definite symmetric matrix, x and y are two nonnull real vectors satisfying such that
where are the ordered eigenvalues of A. Inequality (1) is usually called Wielandt inequality in literature; see Drury et al. [1]. Gustafson [2] gave some meaning of this inequality.
Let the random vector h has the covariance matrix A, then the maximum of the squared correlation is given as follows:
If we set
then the Wielandt inequality (1) becomes the Kantorovich inequality:
Many authors have been studied the Kantorovich inequality, for more details, see Liu [3, 4], Rao and Rao [5] and Liu and Heyde [6].
Wang and Ip [7] have extended the Wielandt inequality to the matrix version, which can be expressed as follows. Suppose X and Y be and matrices satisfying , then
where inequality (5) refers to the Lo\"{w}ner partial ordering.
In inequality (5), A be a positive definite matrix, Lu [8] has extended A to be a nonnegative definite matrix. Drury et al. [1] introduced the matrix, determinant and trace version of the Wielandt inequality. Liu et al. [9] has improved two matrix trace Wielandt inequalities and proposed their statistical applications. Wang and Yang [10] presented the Euclidean norm matrix Wielandt inequality and showed the statistical applications. In this article, we will provide a matrix spectral norm Wielandt inequality and give its application to statistics.
The rest of the article is given as follows. In Section 2, we present a matrix spectral norm versions of the Wielandt inequality. In Section 3, a new measure of association based on the spectral norm is proposed and its upper bound is obtained by using the results in previous section; then we propose an alterative based on the spectral norm of the relative gain of the covariance adjusted estimators of the parameters and its upper bound. Finally, some concluding remarks are given in Section 4.
2 Matrix spectral norm Wielandt inequality
We start this section with some notation. Let be an nonnegative definite matrix of ranka with ; is the nonnegative definite square root of A; X is an matrix of rankk with ; stands for a generalized inverse of a matrix; represents the Moore-Penrose inverse of a matrix; denotes the rank of a matrix; shows for the transpose of a matrix; and stands for the column space of a matrix. Suppose that stands for the orthogonal projectors onto the column space of matrix A, and use the notation
for the orthogonal projector onto .
In order to prove the main results it is necessary to introduce some lemmas.
Lemma 2.1 [4]
Let be an matrix with ranka, and let X be an matrix of rankk satisfying , with . Then
where are the nonzero eigenvalues of A.
Lemma 2.2 [9]
If , and , then
Lemma 2.3 Let be an matrix with ranka, and let X be an matrix of rankk satisfying , with . Then .
Proof As A be a nonnegative definite matrix, we can easily get . By Marsaglia and Styan [11], we have
Since , so we have , then we obtain , that is, , thus
On the other hand
So we get . □
Lemma 2.4 Let be an matrix with ranka, and let X be an matrix of rankk satisfying , with . Then
where denotes the spectral norm of the matrix G, stands for the largest eigenvalues of matrix G, , are the nonzero eigenvalues of A, is the condition number of matrix .
Proof By the definition of the spectral norm, we obtain
By Lemma 2.3, we have , thus we get
So we have
On the other hand, define the condition number of the matrix as , then we have
Thus,
Then, using Lemma 2.1, we get
□
Now we present the first theorem of this article.
Theorem 2.1 Suppose to be an matrix of ranka, and suppose X to be an matrix of rankk, and suppose Y to be an matrix such that and with . Then
where denotes the spectral norm of the matrix G, stands for the largest eigenvalues of matrix G, are the nonzero eigenvalues of A, is the condition number of the matrix .
Proof (1) For (19), using Lemma 2.2 and Lemma 2.4, we obtain
Since and , then . Thus we obtain
Inequality (19) is proved.
(2) For (20), from Lemma 2.2 and Lemma 2.4, we can obtain
The proof of inequality (20) is completed. □
Partition matrix as follows:
where of ranka, of rankk, is and is , .
Now we give another theorem.
Theorem 2.2 Suppose A be an nonnegative definite matrix of ranka partitioned as in (24) and suppose that
then
where are the nonzero eigenvalues of A, is the condition number of matrix .
Proof (1) Since , let the matrix X be and matrix Y be , then we obtain , , , , , , , and . Substituting it into Theorem 2.1, we can get the two inequalities involving .
(2) As , A can be partitioned as in (24) with and , then and . On the other hand, using , we get , which is needed in Theorem 2.1. □
3 Applications to statistics
In this section, we give several inequalities involving covariance matrices, an alternative based on the spectral norm of the relative gain of the covariance adjusted estimator and its upper bound by using the inequalities in Section 2.
3.1 New measure of association
Suppose that μ and ν are and random vectors and that we have the covariance matrix
where .
Wang and Ip [7] have discussed the following measure of association for :
where refers to the determinant of the concerned matrix and . We can see that cannot be used when . As pointed out by Groß [12], the authors may encounter a singular covariance matrix. To solve this problem, Liu et al. [9] introduced a new measure association:
They also gave an upper bound of and they pointed out that is useful in canonical correlations and regression analysis areas as discussed by Lu [8], Wang and Ip [7], and Anderson [13].
Wang and Yang [10] presented an alternative measure association, which is defined as follows:
where stands for the Euclidean norm of concerned matrix and they also gave an upper bound of .
As is well known, there is no measure association involving the spectral norm, so we present a new measure of association based on the spectral norm:
Theorem 3.1 The upper bound of is given as follows:
where are the ordered eigenvalues of Σ, is the condition number of matrix Σ.
Proof It is easy to prove inequality (32) by using Theorem 2.2 and (31). □
3.2 Wishart matrices
Let S be an estimator of Σ, partitioned S as follows:
where is a matrix.
Wang and Ip [7] presented these interesting relations among these submatrices occurring in much of the statistical literature, such as in linear models
where and refer to the largest and smallest eigenvalues of S, respectively. They also considered the concept of the relative gain of the covariance adjusted estimator of a parameter vector discussed by Rao [14] and Wang and Yang [15]. can be regarded as the relative gain and it can be estimated by . Liu et al. [9] use to estimate and they also showed that
where are the ordered eigenvalues of S.
Wang and Yang [7] also studied this problem and used to estimate ; they also gave an upper bound of , which is given as follows:
where and .
In this article we will present the spectral norm operator instead of the determinant, trace, and Euclidean norm. The new relative gain of the covariance adjusted estimator is denoted by
and ω is estimated by
Now we give the upper bound of ω.
Theorem 3.2 The relative gain is bounded as follows:
where are the ordered eigenvalues of S, is the condition number of matrix S.
Proof Using Theorem 2.2, we can easily get the proof of Theorem 3.2. □
Remark 3.1 The result in Theorem 3.2 can be extended to the nonnegative definite matrix , but .
4 Concluding remarks
In this article, we have presented two matrix spectral norm Wielandt inequalities and some applications of the spectral norm Wielandt inequalities, and we also can see that these applications are meaningful, useful, and practical in statistics.
References
Drury SW, Liu S, Lu CY, Puntanen S, Styan GPH: Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments. Sankhya, Ser. A 2002, 64: 453-507.
Gustafson K: The geometrical meaning of the Kantorovich-Wielandt inequalities. Linear Algebra Appl. 1999, 296: 143-151. 10.1016/S0024-3795(99)00106-8
Liu SZ Tinbergen Institute Research Series 106. In Contributions to Matrix Calculus and Applications in Econometrics. Thesis Publishers, Amsterdam; 1995.
Liu SZ: Efficiency comparisons between the OLSE and the BLUE in a singular linear model. J. Stat. Plan. Inference 2000, 84: 191-200. 10.1016/S0378-3758(99)00149-4
Rao CR, Rao MB: Matrix Algebra and Its Applications to Statistics and Econometrics. World Scientific, Singapore; 1998.
Liu SZ, Heyde CC: Some efficiency comparisons for estimators from quasi-likelihood and generalized estimating equations. Lecture Notes-Monograph Series 42. In Mathematical Statistics and Applications: Festschrift for Constance van Eeden. Edited by: Moore M, Froda S, Léger C. Inst. Math. Statist., Beachwood; 2003:357-371.
Wang SG, Ip WC: A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 1999, 296: 171-181. 10.1016/S0024-3795(99)00117-2
Lu, CY: A generalized matrix version of the Wielandt inequality with some applications. Research Report, Department of Mathematics, North east Normal University, Changchun, China, 8 pp. (1999)
Liu SZ, Lu CY, Puntanen S: Matrix trace Wielandt inequalities with statistical applications. J. Stat. Plan. Inference 2009, 139: 2254-2260. 10.1016/j.jspi.2008.10.026
Wang LT, Yang H: Matrix Euclidean norm Wielandt inequalities and their applications to statistics. Stat. Pap. 2012, 53: 521-530. 10.1007/s00362-010-0357-y
Marsaglia G, Styan GPH: Equalities and inequalities of ranks of matrices. Linear Multilinear Algebra 1974, 2: 269-292. 10.1080/03081087408817070
Groß J: The general Gauss-Markov model with possibly singular dispersion matrix. Stat. Pap. 2004, 45: 311-336. 10.1007/BF02777575
Anderson TW: An Introduction to Multivariate Statistical Analysis. 3rd edition. Wiley, New York; 2003.
Rao CR: Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965-66), vol. I: Statistics. Edited by: Cam LM, Neyman J. University of California Press, Berkeley; 1967:355-372.
Wang SG, Yang ZH: Pitman optimality of covariance-improved estimators. Chin. Sci. Bull. 1995, 40: 1150-1154.
Acknowledgements
The authors are grateful to the editor and the two anonymous referees for their valuable comments which improved the quality of the paper. This work was supported by the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grant No: R2013SC12), the National Natural Science Foundation of China (Grant Nos: 71271227, 11201505), and Program for Innovation Team Building at Institutions of Higher Education in Chongqing (Grant No: KJTD201321).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wu, J., Yi, W. Matrix spectral norm Wielandt inequalities with statistical applications. J Inequal Appl 2014, 110 (2014). https://doi.org/10.1186/1029-242X-2014-110
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-110