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Inequalities for ranks of matrix expressions involving generalized inverses
Journal of Inequalities and Applications volume 2014, Article number: 87 (2014)
Abstract
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.
MSC:15A09, 15A24.
1 Introduction
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 generalized inverse X of A is a matrix which satisfies some of the following four Penrose equations [1]:
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 [5], 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 [8] and Marsaglia [9], gives upper and lower bounds for the rank of the sum of two matrices.
Given a matrix expression with some variant matrices in it, the rank of the matrix expression will vary with respect to the variant matrices. Since the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix [10], then the inequalities for ranks of matrix expressions must exist with respect to their variant matrices. Many problems in matrix theory and applications are closely related to the inequalities for ranks of matrix expressions with variant matrices. For example, a matrix expression of order n is nonsingular if and only if the maximal rank of with respect to X is n; a matrix equation is consistent if and only if the minimal rank of the matrix expression with respect to X is zero; two consistent matrix equations and have a common solution if and only if the minimal rank of the difference of their solutions is zero. In general, for any two matrix expressions and of the same size, there are and such that if and only if
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.
Let , , and . Then
where
Lemma 1.2 [20]
Suppose B, C and D satisfy and . Then the Moore-Penrose inverse of the block matrix
can be expressed as
Lemma 1.3 [19]
Let , , and . Then
-
(I)
,
-
(II)
,
-
(III)
,
-
(IV)
,
-
(V)
,
-
(VI)
,
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.
Theorem 2.1 Let , and . Then for any and , the following inequalities hold:
Proof Using formula (2) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
The last equation holds, since
i.e. .
Using formula (2) in Lemma 1.1 again, we have
The third equation holds, since from formula (III) in Lemma 1.3,
Combining (10) with (11), we have
That is, for any and , the following inequalities hold:
On the other hand, applying formula (3) in Lemma 1.1, we have
According to (13), we have
By formula (3) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
Using formula (2) in Lemma 1.1 and formulas (III), (IV) in Lemma 1.3, we have
Combining the formulas (14), (15) with (16), we obtain
That is, for any and , the following inequality holds:
 □
Substituting in Theorem 2.1 with , we immediately obtain the following corollaries.
Corollary 2.1 ([[21], Theorem 2.2], [[22], Theorem 2.3])
For any matrices and , the identity holds for any and if and only if
In particular, substituting in Theorem 2.1 with leads to the following result.
Corollary 2.2 Let and . Then for any and , the following inequalities holds:
Corollary 2.3 Let and . Then the identity
holds, for any and , if and only if
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.
Theorem 3.1 Let , and . Then for any and , the following inequalities hold:
where
Proof Applying formula (4) in Lemma 1.1 and the formulas (III), (IV) in Lemma 1.3, we have
The second equation holds, since , and
Applying the formulas (4) in Lemma 1.1 again, we have
The third equation holds, since from formula (III) in Lemma 1.3
and
In view of (18) and (19) it follows that
That is, for any and , the following inequalities hold:
On the other hand, using formula (5) in Lemma 1.1 and formula (I) in Lemma 1.3, we have
where
and
According to the results in (21), we have
Note that, for any ,
and
Applying formula (5) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
and
Combining (23), (25) with (26), we have
That is,
According to formula (4) in Lemma 1.1 and the formulas (III), (IV) in Lemma 1.3, we have
By (24) and (28), we have
That is,
Finally on account of (22), (27), and (29), it is seen that
That is, for any and , we have
 □
From Theorem 3.1, we immediately obtain the following corollaries by the formulas (20) and (30).
Corollary 3.1 Let and . Then the identity
holds for any and if and only if
In particular, substituting in Theorem 3.1 with leads to the following.
Corollary 3.2 Let and . Then for any and , the following inequalities holds:
Corollary 3.3 Let and . Then the identity
holds for any and if and only if
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.
Theorem 4.1 Let , and . Then for any and , the following inequalities hold:
Proof According to formula (6) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
Applying formula (6) in Lemma 1.1 and formula (IV) in Lemma 1.3 again, we have
The third equation holds, since from formula (III) in Lemma 1.3,
Combining (31) with (32), we have
That is, for any and , the following inequalities hold:
On the other hand, from formula (7) in Lemma 1.1 and formula (IV) in Lemma 1.3, we have
From (34), we get
Applying formula (7) in Lemma 1.1 and formula (IV) in Lemma 1.3 again, we have
Applying formula (6) in Lemma 1.1 and the formulas (I), (III) in Lemma 1.3, we have
The third equation holds, since
By (35), (36), and (37), we have
That is, for any and , we have
 □
From Theorem 4.1, we immediately obtain the following corollaries by (33) and (38).
Corollary 4.1 Let and . Then the identity
holds for any and if and only if
Corollary 4.2 Let and . Then for any and , the following inequalities hold:
Corollary 4.3 Let and . Then the identity
holds for any and if and only if
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.
Theorem 4.2 Let , and . Then for any and , the following inequalities hold:
Corollary 4.4 Let and . Then the identity
holds for any and if and only if
Corollary 4.5 Let and . Then for any and , the following inequalities hold:
Corollary 4.6 Let and . Then the identity
holds for any and if and only if
5 Inequalities for ranks of and
From the results in [3], it is seen that a matrix X belongs to if and only if
where is arbitrary. Similarly, a matrix X belongs to if and only if
where is arbitrary. Thus
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.
Theorem 5.1 Let , and . Then for any and , the following inequalities hold:
Theorem 5.2 Let , and . Then for any and , the following inequalities hold:
6 Rank of
In this section, we will present the rank of the linear matrix expression
where , and are given matrices.
Theorem 6.1 Let , and . Then
Proof Let
Then applying formula (9) in Lemma 1.2, we have
The sub-matrix in the upper left corner of the Moore-Penrose inverse of T can be expressed as
where and . Hence
Applying formula (8) in Lemma 1.1, we have
 □
As a direct consequence of Theorem 6.1, we immediately get the following results.
Corollary 6.1 Let and . Then the identity holds if and only if
Corollary 6.2 Let and . Then the identity holds if and only if
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Acknowledgements
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.
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Xiong, Z. Inequalities for ranks of matrix expressions involving generalized inverses. J Inequal Appl 2014, 87 (2014). https://doi.org/10.1186/1029-242X-2014-87
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DOI: https://doi.org/10.1186/1029-242X-2014-87
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