Inequalities for ranks of matrix expressions involving generalized inverses
© Xiong; licensee Springer. 2014
Received: 18 September 2013
Accepted: 10 February 2014
Published: 20 February 2014
In this paper, we present several inequalities for ranks of the matrix expressions with respect to the choice of X, where X is taken, respectively, as , , , , as well as and . Various application of these inequalities are also presented.
Keywordsmatrix expression generalized inverse inequality rank generalized Schur complement
Throughout this paper denotes the set of all matrices over the complex field C. denotes the identity matrix of order k, is the matrix of all zero entries (if no confusion occurs, we will drop the subscript). For a matrix , , , and denote the conjugate transpose, the range space, the null space, and the rank of the matrix A, respectively.
For a subset of the set , the set of matrices satisfying the equations from among equations is denoted by . Arbitrary matrix from is called an -inverse of A and is denoted by . For example, an matrix X of the set is called a g-inverse of A and is denoted by . The well-known seven common types of generalized inverses of A introduced from (1) are, respectively, the -inverse (g-inverse), -inverse (reflexive g-inverse), -inverse (least square g-inverse), -inverse (minimum norm g-inverse), -inverse, -inverse and -inverse. The unique -inverse of A is denoted by , which is called the Moore-Penrose inverse of A. For convenience, the symbols and stand for the two orthogonal projectors and . We refer the reader to [2–4] for basic results on generalized inverses.
The notion of rank of a matrix appears to have been introduced , by Sylvester in 1851. Two principal classical results [6, 7] on rank are Sylvester’s law of nullity and Frobenius’ inequality. A modern inequality, obtained by Khatri  and Marsaglia , gives upper and lower bounds for the rank of the sum of two matrices.
The inequalities for ranks of a matrix expression play the important roles in matrix theory for describing the dimension of the row and column vector space of the matrix expressions, which are well understand and are easy to compute by the well-known elementary or congruent matrix expressions, see, e.g., [8, 11–15]. The inequalities for ranks of matrix expressions could be regarded as one of the fundamental topics in matrix theory and applications, which can be used to investigate nonsingularity and inverse of a matrix, range and rank invariance of a matrix, relations between subspaces, equalities of matrix expressions with variable matrices, reverse order laws for generalized inverses, existence of solutions to various matrix equations, and so on, see, e.g., [8, 9, 16–19].
In this paper, by using the maximal and minimal ranks of generalized Schur complement [11, 14], we get several inequalities for ranks of the matrix expressions , where X is taken, respectively, as , , , , , and . We also derive various valuable consequences.
In order to find the inequalities for ranks of matrix expressions, we first mention the following lemmas, which will be used in this paper.
Lemma 1.2 
Lemma 1.3 
where , , and dim denotes dimension.
2 Inequalities for ranks of
In this section, we will present several inequalities for ranks of the matrix expression , with respect to two variant matrices and , where , and are given matrices.
Substituting in Theorem 2.1 with , we immediately obtain the following corollaries.
In particular, substituting in Theorem 2.1 with leads to the following result.
3 Inequalities for ranks of
By analogy with the proof of Theorem 2.1, in this section we will present several inequalities for ranks of the matrix expression . The main result in this section is the following theorem.
From Theorem 3.1, we immediately obtain the following corollaries by the formulas (20) and (30).
In particular, substituting in Theorem 3.1 with leads to the following.
4 Inequalities for ranks of and
Applying the formulas (6) and (7) in Lemma 1.1 to the matrix expressions and , we obtain some inequalities for ranks of this two matrix expressions. The main result in this section is the following theorem.
From Theorem 4.1, we immediately obtain the following corollaries by (33) and (38).
Notice that a matrix X belongs to if and only if belongs to . So from the results obtained in above of Section 4, we can get the inequalities for ranks of . We state the following theorems without proofs.
5 Inequalities for ranks of and
Applying the formulas (2) and (3) in Lemma 1.1 to (39) and (40), we have the following theorems, which can be shown by a similar approach to Theorem 2.1, and the proof are omitted here.
6 Rank of
where , and are given matrices.
As a direct consequence of Theorem 6.1, we immediately get the following results.
The author would like to thank the Editor-in-Chief and the anonymous referees for their very detailed comments, which greatly improved the presentation of this article. The work was supported by the NSFC (Grant No: 11301397) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2012LYM-0126) and the Basic Theory and Scientific Research of Science and Technology Project of Jiangmen City, China, 2013.
- Penrose R: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 1955, 51: 406-413. 10.1017/S0305004100030401MATHMathSciNetView ArticleGoogle Scholar
- Ben-Israel A, Greville TNE: Generalized Inverse: Theory and Applications. Wiley-Interscience, New York; 1974. 2nd edn. Springer, New York (2002)MATHGoogle Scholar
- Rao CR, Mitra SK: Generalized Inverse of Matrices and Its Applications. Wiley, New York; 1971.MATHGoogle Scholar
- Wang G, Wei Y, Qiao S: Generalized Inverses: Theory and Computations. Science Press, Beijing; 2004.Google Scholar
- Mirsky L: An Introduction to Linear Algebra. Oxford University Press, Oxford; 1955.MATHGoogle Scholar
- Frobenius G III. In Über den rang einer matrix. Springer, Berlin; 1968:479-490.Google Scholar
- Sylvester JJ IV. In The Collected Mathematical Papers of James Joseph Sylvester. Cambridge University Press, Cambridge; 1912:133-145.Google Scholar
- Khatri CG: A simplified approach to the derivation of the theorems on the rank of a matrix. J. Maharaja Sayajirao Univ. Baroda 1961, 10: 1-5.Google Scholar
- Marsaglia G: Bounds on the rank of the sum of matrices. In Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes. Academia, Prague; 1967:455-462.Google Scholar
- David CL: Linear Algebra and Its Applications. Addison-Wesley, Reading; 1994.Google Scholar
- Tian Y: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 2002, 355: 187-214. 10.1016/S0024-3795(02)00345-2MATHMathSciNetView ArticleGoogle Scholar
- Tian Y:Ranks of solutions of the matrix equation . Linear Multilinear Algebra 2003, 51: 111-125. 10.1080/0308108031000114631MathSciNetView ArticleGoogle Scholar
- Tian Y, Cheng S:The maximal and minimal ranks of with applications. N.Y. J. Math. 2003, 9: 345-362.MATHMathSciNetGoogle Scholar
- Tian Y: More on maximal and minimal ranks of Schur complements with applications. Appl. Math. Comput. 2004, 152: 675-692. 10.1016/S0096-3003(03)00585-XMATHMathSciNetView ArticleGoogle Scholar
- Tian Y: The maximal and minimal ranks of a quadratic matrix expression with applications. Linear Multilinear Algebra 2011, 59: 627-644. 10.1080/03081081003774268MATHMathSciNetView ArticleGoogle Scholar
- Groß J: Comment on range invariance of matrix products. Linear Multilinear Algebra 1996, 41: 157-160. 10.1080/03081089608818469MATHView ArticleGoogle Scholar
- Groß J, Tian Y: Invariance properties of a triple matrix product involving generalized inverses. Linear Algebra Appl. 2006, 417: 94-107. 10.1016/j.laa.2006.03.026MATHMathSciNetView ArticleGoogle Scholar
- Johnson CR, Whitney GT: Minimum rank completions. Linear Multilinear Algebra 1991, 28: 271-273. 10.1080/03081089108818051MATHMathSciNetView ArticleGoogle Scholar
- Marsaglia G, Styan GPH: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 1974, 2: 269-292. 10.1080/03081087408817070MathSciNetView ArticleGoogle Scholar
- Hartwig RE: Block generalized inverses. Arch. Ration. Mech. Anal. 1976, 61: 197-251. 10.1007/BF00281485MATHMathSciNetView ArticleGoogle Scholar
- Wei M: Equivalent conditions for generalized inverses of products. Linear Algebra Appl. 1997, 266: 347-363.MATHMathSciNetView ArticleGoogle Scholar
- Werner HJ: When is a generalized inverse of AB ? Linear Algebra Appl. 1994, 210: 255-263.MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.