ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES

Transcription

1 olume , Issue 2, Article 41, 10 pp. ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES HANYU LI, HU YANG, AND HUA SHAO COLLEGE OF MATHEMATICS AND PHYSICS CHONGQING UNIERSITY CHONGQING, , P.R. CHINA DEPARTMENT OF MATHEMATICS & PHYSICS CHONGQING UNIERSITY OF SCIENCE AND TECHNOLOGY CHONGQING, , P.R. CHINA Received 21 January, 2008; accepted 19 May, 2009 Communicated by S.S. Dragomir ABSTRACT. In this paper, the relations between the weighted partial orderings on the set of rectangular complex matrices are first studied. Then, using the matrix function defined by Yang and Li [H. Yang and H.Y. LI, Weighted UD -decomposition and weighted spectral decomposition for rectangular matrices and their applications, Appl. Math. Comput , pp ], some weighted partial orderings of matrices are compared with the orderings of their functions. Key words and phrases: Weighted partial ordering, Matrix function, Singular value decomposition Mathematics Subject Classification. 15A45, 15A INTRODUCTION Let C m n denote the set of m n complex matrices, C m n r denote a subset of C m n comprising matrices with rank r, C m denote a set of Hermitian positive semidefinite matrices of order m, and C m > denote a subset of C m consisting of positive definite matrices. Let I r be the identity matrix of order r. Given A C m n, the symbols A, A # MN, RA, and ra stand for the conjugate transpose, weighted conjugate transpose, range, and rank, respectively, of A. Details for the concept of A # MN can be found in [11, 13]. Moreover, unless otherwise specified, in this paper we always assume that the given weight matrices M C m m and N C n n. In the following, we give some definitions of matrix partial orderings. Definition 1.1. For A, B C m m, we say that A is below B with respect to: The authors would like to thank the editors and referees for their valuable comments and helpful suggestions, which improved the presentation of this paper

2 2 HANYU LI, HU YANG, AND HUA SHAO 1 the Löwner partial ordering and write A L B, whenever B A C m. 2 the weighted Löwner partial ordering and write A W L B, whenever MB A C m. Definition 1.2. For A, B C m n, we say that A is below B with respect to: 1 the star partial ordering and write A B, whenever A A = A B and AA = BA. 2 the weighted star partial ordering and write A # B, whenever A # MN A = A# MN B and AA # MN = BA# MN. 3 the W G-weighted star partial ordering and write A # W G B, whenever MAB # MN C m, NA # MN B Cn, and AA # MN W L AB # MN. 4 the W GL partial ordering and write A W GL B, whenever AA # MN 1/2 W L BB # MN 1/2 and AB # MN = AA# MN 1/2 BB # MN 1/2. 5 the W GL2 partial ordering and write A W GL2 B, whenever AA # MN W L BB # MN and AB # MN = AA# MN 1/2 BB # MN 1/2. 6 the minus partial ordering and write A B, whenever A A = A B and AA = = BA = for some possibly distinct generalized inverses A, A = of A satisfying AA A = A = AA = A. The weighted Löwner and weighted star partial orderings can be found in [6, 15] and [9], respectively. The W GL partial ordering was defined by Yang and Li in [15] and the W GL2 partial ordering can be defined similarly. The minus partial ordering was introduced by Hartwig [2], who also showed that the minus partial ordering is equivalent to rank subtractivity, namely A B if and only if rb A = rb ra. For the relation # W G, we can use Lemma 2.5 introduced below to verify that it is indeed a matrix partial ordering according to the three laws of matrix partial orderings. Baksalary and Pukelsheim showed how the partial orderings of two Hermitian positive semidefinite matrices A and B relate to the orderings of their squares A 2 and B 2 in the sense of the Löwner partial ordering, minus partial ordering, and star partial ordering in [1]. In terms of these steps, Hauke and Markiewicz [3] discussed how the partial orderings of two rectangular matrices A and B relate to the orderings of their generalized square A 2 and B 2, A 2 = AA A 1/2, in the sense of the GL partial ordering, minus partial ordering, G-star partial ordering, and star partial ordering. The definitions of the GL and G-star partial orderings can be found in [3, 4]. In addition, Hauke and Markiewicz [5] also compared the star partial ordering A B, G-star partial ordering A G B, and GL partial ordering A GL B with the orderings fa fb, fa G fb, and fa GL fb, respectively. Here, fa is a matrix function defined in A [7]. Legiša [8] also discussed the star partial ordering and surjective mappings on C n n. These results extended the work of Mathias [10] to some extent, who studied the relations between the Löwner partial ordering A L B and the ordering fa L fb. In the present paper, based on the definition A 2 = AA # MN A1/2 also called the generalized square of A, we study how the partial orderings of two rectangular matrices A and B relate to the orderings of their generalized squares A 2 and B 2 in the sense of the W GL partial ordering, W G-weighted star partial ordering, weighted star partial ordering, and minus partial ordering. Further, adopting the matrix functions presented in [14], we also compare the weighted partial orderings A # B, A # W G B, and A W GL B with the orderings fa # fb, fa # W G fb, and fa W GL fb, respectively. These works generalize the results of Hauke and Markiewicz [3, 5].

3 WEIGHTED PARTIAL ORDERINGS 3 Now we introduce the M, N weighted singular value decomposition [11, 12] MN-SD and the matrix functions based on the MN-SD, which are useful in this paper, Lemma 1.1. Let A C m n r. Then there exist U C m m and C n n satisfying U MU = I m and N 1 = I n such that 1.1 A = U D 0, where D = diagσ 1,..., σ r, σ i = λ i > 0, and λ 1 λ r > 0 are the nonzero eigenvalues of A # MN A = N 1 A MA. Here, σ 1 σ r > 0 are called the nonzero M, N weighted singular values of A. If, in addition, we let U = U 1, U 2 and = 1, 2, where U 1 C m r and 1 C n r, then 1.2 U1 MU 1 = 1 N 1 1 = I r, A = U 1 D1. Considering the MN-SD, from [14], we can rewrite the matrix function fa : C m n C m n by way of fa = U 1 fd1 using the real function f, where fd is the diagonal matrix with diagonal elements fσ 1,..., fσ r. More information on the matrix function can be found in [14]. 2. RELATIONS BETWEEN THE WEIGHTED PARTIAL ORDERINGS Firstly, it is easy to obtain that on the cone of generalized Hermitian positive semidefinite matrices namely the cone comprising all matrixes which multiplied by a given Hermitian positive definite matrix become Hermitian positive semidefinite matrices the W GL partial ordering coincides with the weighted Löwner partial ordering, i.e., for matrices A, B C m m satisfying MA, MB C m, A W GL B if and only if A W L B and the W GL2 partial ordering coincides with the W GL partial ordering of the squares of matrices, i.e., for matrices A, B C mm satisfying MA, MB C m, A W GL2 B if and only if A 2 W GL B 2. On the set of rectangular matrices, for the generalized square of A, i.e., A 2 = AA # MN A1/2, the above relation takes the form: 2.1 A W GL2 B if and only if A 2 W GL B 2, which will be proved in the following theorem. Theorem 2.1. Let A, B C m n, ra = a, and rb = b. Then 2.1 holds. Proof. It is easy to find that the first conditions in the definitions of W GL2 partial ordering for A and B and W GL partial ordering for A 2 and B 2 are equivalent. To prove the equivalence of the second conditions, let us use the MN-SD introduced in Lemma 1.1. Let A = U 1 D a 1 and B = U 2 D b 2 be the MN-SDs of A and B, where U 1 C m a, U 2 C m b, 1 C n a, and 2 C n b satisfying U1 MU 1 = 1 N 1 1 = I a and U2 MU 2 = 2 N 1 2 = I b, and D a C a >, D b C b > are diagonal matrices. Then 2.2 AB # MN =AA# MN 1/2 BB # MN 1/2 U 1 D a 1 N 1 2 D b U 2 M = U 1 D a 1 N 1 1 D a U 1 M 1/2 U 2 D b 2 N 1 2 D b U 2 M 1/2 U 1 D a 1 N 1 2 D b U 2 M = U 1 D a U 1 MU 2 D b U 2 M 1 N 1 2 = U 1 MU 2.

4 4 HANYU LI, HU YANG, AND HUA SHAO Note that 2.3 Similarly, A 2 = AA # MN A1/2 = U 1 D a 1 N 1 1 D a U 1 MU 1 D a 1 1/2 = U 1 D a 1 N 1 1 D a 1 = U 1 D 2 a B 2 = U 2 D 2 b 2. Then A 2 B 2 # MN =A2 A 2 # MN 1/2 B 2 B 2 # MN 1/2 which together with 2.2 gives U 1 D 2 a 1 N 1 2 D 2 bu 2 M = U 1 D 2 a 1 N 1 1 D 2 au 1 M 1/2 U 2 D 2 b 2 N 1 2 D 2 bu 2 M 1/2 U 1 D 2 a 1 N 1 2 D 2 bu 2 M = U 1 D 2 au 1 MU 2 D 2 bu 2 M 1 N 1 2 = U 1 MU 2, AB # MN = AA# MN 1/2 BB # MN 1/2 A 2 B 2 # MN = A2 A 2 # MN 1/2 B 2 B 2 # MN 1/2. Therefore, the proof is completed. Before studying the relation between the W GL partial orderings for A and B and that for their generalized squares, we first introduce a lemma from [1]. Lemma 2.2. Let A, B C m. Then a If A 2 L B 2, then A L B. b If AB = BA and A L B, then A 2 L B 2. Theorem 2.3. Let A, B C m n, ra = a, rb = b, and a A W GL B, b A 2 W GL B 2, c AB # MN # MM = AB# MN. Then b implies a, and a and c imply b. Proof. i. b a. Together with Theorem 2.1 and the definitions of W GL2 and W GL partial orderings, it suffices to show that 2.5 A 2 A 2 # MN 1/2 W L B 2 B 2 # MN 1/2 AA # MN 1/2 W L BB # MN 1/2. From the proof of Theorem 2.1 and the definition of weighted Löwner partial ordering, we have 2.6 A 2 A 2 # MN 1/2 W L B 2 B 2 # MN 1/2 U 1 DaU 2 1 M W L U 2 DbU 2 2 M MU 1 DaU 2 1 M L MU 2 DbU 2 2 M M 1/2 U 1 DaU 2 1 M 1/2 L M 1/2 U 2 DbU 2 2 M 1/2 M 1/2 U 1 D a U1 M 1/2 M 1/2 U 1 D a U1 M 1/2 L M 1/2 U 2 D b U2 M 1/2 M 1/2 U 2 D b U2 M 1/2.

5 WEIGHTED PARTIAL ORDERINGS 5 Applying Lemma 2.2 a to 2.6 leads to 2.7 M 1/2 U 1 D a U 1 M 1/2 L M 1/2 U 2 D b U 2 M 1/2 Then, by 2.6 and 2.7, we show that 2.5 holds. MU 1 D a U 1 M L MU 2 D b U 2 M MAA # MN 1/2 L MBB # MN 1/2 AA # MN 1/2 W L BB # MN 1/2. ii. a and c b. Similarly, combining with Theorem 2.1 and the definitions of W GL2 and W GL partial orderings, we only need to prove that 2.8 AA # MN 1/2 W L BB # MN 1/2 A 2 A 2 # MN 1/2 W L B 2 B 2 # MN 1/2. From the proof of Theorem 2.1 and the definition of weighted Löwner partial orderings, we have 2.9 According to c, we have AA # MN 1/2 W L BB # MN 1/2 U 1 D a U 1 M W L U 2 D b U 2 M MU 1 D a U 1 M L MU 2 D b U 2 M M 1/2 U 1 D a U 1 M 1/2 L M 1/2 U 2 D b U 2 M 1/ U 2 D b 2 N 1 1 D a U 1 M = U 1 D a 1 N 1 2 D b U 2 M. Thus, together with 2.10 and 2.2, we can obtain 2.11 U 2 D b U 2 MU 1 D a U 1 M = U 1 D a U 1 MU 2 D b U 2 M M 1/2 U 1 D a U 1 M 1/2 M 1/2 U 2 D b U 2 M 1/2 Applying Lemma 2.2 b to 2.11 and 2.9, we have = M 1/2 U 2 D b U 2 M 1/2 M 1/2 U 1 D a U 1 M 1/ M 1/2 U 1 D a U 1 M 1/2 M 1/2 U 1 D a U 1 M 1/2 L M 1/2 U 2 D b U 2 M 1/2 M 1/2 U 2 D b U 2 M 1/2. Then, combining with 2.12 and 2.6, we can show that 2.8 holds. The weighted star partial ordering was characterized by Liu in [9], using the simultaneous weighted singular value decomposition of matrices [9]. He obtained the following result. Lemma 2.4. Let A, B C m n and rb = b > ra = a 1. Then A # B if and only if there exist matrices U C m m and C n n satisfying U MU = I m and N 1 = I n such that Da 0 A = U = U 1 D a 1, B = U D a 0 D 0 0 = U 2 Da 0 0 D 2, where U 1 C m a, 1 C n a and U 2 C m b, 2 C n b denote the first a and b columns of U,, respectively, and satisfy U1 MU 1 = 1 N 1 1 = I a and U2 MU 2 = 2 N 1 2 = I b, and D a C a > and D C b a > are diagonal matrices.

6 6 HANYU LI, HU YANG, AND HUA SHAO Similarly to Lemma 2.4, we can take the following form to characterize the W G-weighted star partial ordering. A detailed proof is omitted. Lemma 2.5. Let A, B C m n and rb = b > ra = a 1. Then A # W G B if and only if there exist matrices U C m m and C n n satisfying U MU = I m and N 1 = I n such that Da 0 A = U = U 1 D a 1, B = U D a 0 D 0 0 = U 2 Da 0 0 D 2, where U 1 C m a, 1 C n a and U 2 C m b, 2 C n b denote the first a and b columns of U,, respectively, and satisfy U1 MU 1 = 1 N 1 1 = I a and U2 MU 2 = 2 N 1 2 = I b, and D a, D a C a > and D C b a > are diagonal matrices, and D a D a C a. From the simultaneous weighted singular value decomposition of matrices [9], Lemma 2.4, and Lemma 2.5, we can derive the following theorem. Theorem 2.6. Let A, B C m n. Then a A # B MAB # MN Cm, NA # MN B Cn, and AA # MN = AA# MN 1/2 BB # MN 1/2. b A # W G B MAB # MN Cm, NA # MN B Cn, and AA # MN 1/2 W L BB # MN 1/2. Considering Definition 1.24 and Theorem 2.6, we can present the following relations between three weighted partial orderings by the sequence of implications: A # B A # W G B A W GL B. As in Theorem 2.3, we now discuss the corresponding result for W G-weighted star partial ordering using Lemma 2.5. Theorem 2.7. Let A, B C m n, ra = a, and rb = b. Then A 2 # W G B 2 if and only if A # W G B. Proof. Let the MN-SDs of A and B be as in the proof of Theorem 2.1. Considering Lemma 1.1, from 2.3, 2.4, and Lemma 2.5, we have D A 2 = U 1 Da = U a 0, D B 2 = U 2 Db = U b 0. In this case, the MN-SDs of A and B can be rewritten as Da 0 A = U Db 0, B = U Thus, from Lemma 2.5, we have A 2 # W G B 2 A # W G B..

7 WEIGHTED PARTIAL ORDERINGS 7 Conversely, from Lemma 2.5, A # W G B is equivalent to Da 0 A = U, B = U D a 0 D 0 0 Then A 2 = U D 2 a 0, B 2 = U Therefore, from Lemma 2.5 again, the proof is completed. D2 b 0 D The characterization of the weighted star partial ordering can be obtained similarly using Lemma 2.4, and is given in the following theorem. Theorem 2.8. Let A, B C m n, ra = a, and rb = b. Then A 2 # B 2 if and only if A # B. The following result was presented by Liu [9]. It is useful for studying the relation between the minus ordering for A and B and that for A 2 and B 2. Lemma 2.9. Let A, B C m n. Then A # B if and only if A B, AB # MN # MM = AB# MN, and A# MN B# NN = A# MN B. Theorem Let A, B C m n, ra = a, rb = b, AB # MN # MM A # MN B# NN = A# MNB. Then = AB# MN, and A 2 B 2 if and only if A B. Proof. According to AB # MN # MM = AB# MN, A# MN B# NN = A# MNB, the proof of Theorem of [9], and the simultaneous unitary equivalence theorem [7], we have A = U Ec 0, B = U Fc 0, where U C m m and C n n satisfy U MU = I m and N 1 = I n, and E c C c c F c are real diagonal matrices, c = max{a, b}. As in 2.3 and 2.4, we can obtain E A 2 2 = U c 0 Thus, it is easy to verify that As a result,, B 2 = U Fc F c 0 A 2 B 2 # MN # MM = A2 B 2 # MN and A 2 # MN B2 # NN = A2 # MN B2. A 2 B 2 A 2 # B 2. By Theorem 2.8 and Lemma 2.9, the proof is completed.. and

8 8 HANYU LI, HU YANG, AND HUA SHAO 3. WEIGHTED MATRIX PARTIAL ORDERINGS AND MATRIX FUNCTIONS In this section, we study the relations between some weighted partial orderings of matrices and the orderings of their functions. Here, we are interested in such matrix functions for which r[fa] = ra, i.e., functions for which fx = 0 only for x = 0. These functions are said to be nondegenerating. The following properties of f gathered in Lemma 3.1 will be used in subsequent parts of this section. Lemma 3.1. Let A, B C m n and let f be a nondegenerating matrix function. Then a RA = RfA. b AB # MN = AA# MN 1/2 BB # MN 1/2 fafb # MN = faa# MN 1/2 fbb # MN 1/2. Proof. a. From the MN-SD of A, i.e., 1.2, and the property of f, we have RA = RU 1 D1 = RU 1 = RU 1 fd1 = RfA. b. Similar to the proof of Theorem 2.1, let A = U 1 D a 1 and B = U 2 D b 2 be the MN-SDs of A and B respectively. Considering the definition of matrix functions, we obtain fafb # MN =faa# MN 1/2 fbb # MN 1/2 U 1 fd a 1 N 1 2 fd b U 2 M = U 1 fd a U 1 MU 2 fd b U 2 M 1 N 1 2 = U 1 MU 2, which together with 2.2 implies the proof. In the following theorems, we compare some weighted partial orderings of matrices with orderings of their functions. Theorem 3.2. Let A, B C m n and let f be a positive one-to-one function. Then A # B if and only if fa # fb. Proof. From Definition 1.22 and Lemma 2.4, we have that A # B is equivalent to AB # MN = U 1D 2 au 1 M = AA # MN and A# MN B = N 1 1 D 2 a 1 = A # MN A, and fa # fb is equivalent to fafb # MN = U 1fD a 2 U 1 M = fafa # MN and fa # MN fb = N 1 1 fd a 2 1 = fa # MN fa. Then, using the properties of f, the proof is completed. Theorem 3.3. Let A, B C m n and let f be a positive strictly increasing function. Then A # W G B if and only if fa # W G fb. Proof. From Definition 1.12, Definition 1.23, and Lemma 2.5, we obtain that A # W G B is equivalent to MAA # MN = MU 1D 2 au 1 M L MU 1 D a D a U 1 M = MAB # MN, and MAB # MN = MU 1D a D a U 1 M C m, NA # MN B = 1D a D a 1 C n ;

9 WEIGHTED PARTIAL ORDERINGS 9 and fa # W G fb is equivalent to MfAfA # MN = MU 1fD a 2 U 1 M L MU 1 fd a fd a U 1 M = MfAfB # MN, MfAfB # MN = MU 1fD a fd a U1 M C m and NfA # MN B = 1fD a fd a 1 C n. Therefore, the proof follows from the property of f. We need to point out that the above results are not valid for the W GL partial ordering or for the weighted Löwner partial ordering. However, it is possible to reduce the problem of comparing the W GL partial ordering of matrices and the W GL partial ordering of their functions to a suitable problem involving the weighted Löwner partial ordering. Thus, from Definition 1.12, Definition 1.24, and Lemma 3.1, we can deduce the following theorem. Theorem 3.4. Let A, B C m n and let f be a positive strictly increasing function. The following statements are equivalent: a A W GL B if and only if fa W GL fb. b AA # MN 1/2 W L BB # MN 1/2 if and only if faa # MN 1/2 W L faa # MN 1/2. Remark 1. It is worthwhile to note that some of the results of Section 3 can be regarded as generalizations of those in Section 2. For example, if ft = t 2, then fa = U 1 D 2 1 = A 2, hence, in this case, Theorem 3.2 and Theorem 3.3 will reduce to Theorem 2.8 and Theorem 2.7, respectively. REFERENCES [1] J.K. BAKSALARY AND F. PUKELSHEIM, On the Löwner, minus, and star partial orderings of nonnegative definite matrices and their squares, Linear Algebra Appl., , [2] R.E. HARTWIG, How to partially order regular elements?, Math. Japon., , [3] J. HAUKE AND A. MARKIEWICZ, Remarks on partial orderings on the set of rectangular matrices, Discuss. Math. Algebra Stochastic Methods, , [4] J. HAUKE AND A. MARKIEWICZ, On partial orderings on the set of rectangular matrices, Linear Algebra Appl., , [5] J. HAUKE AND A. MARKIEWICZ, On partial orderings on the set of rectangular matrices and their properties, Discuss. Math. Algebra Stochastic Methods, , [6] J. HAUKE AND A. MARKIEWICZ, On orderings induced by the Löwner partial ordering, Appl. Math. Warsaw, , [7] R.A. HORN AND C.R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [8] P. LEGIŠA, Automorphisms of M n, partially ordered by the star order, Linear Multilinear Algebra, , [9] X.J. LIU, Partial orderings and generalized inverses of matrices, PhD Thesis, Xidian University, Xi an, China 2003 in Chinese. [10] R. MATHIAS, The equivalence of two partial orders on a convex cone of positive semidefinite matrices, Linear Algebra Appl., ,

10 10 HANYU LI, HU YANG, AND HUA SHAO [11] C.R. RAO AND S.K. MITRA, Generalized Inverses of Matrices and its Applications, Wiley, New York, [12] C.F. AN LOAN, Generalizing the singular value decomposition, SIAM J. Numer. Anal., , [13] G.R. WANG, Y.M. WEI AND S.Z. QIAO, Generalized Inverses: Theory and Computations, Science Press, Beijing, [14] H. YANG AND H.Y. LI, Weighted UD -decomposition and weighted spectral decomposition for rectangular matrices and their applications, Appl. Math. Comput., , [15] H. YANG AND H.Y. LI, Weighted polar decomposition and WGL partial ordering of rectangular complex matrices, SIAM J. Matrix Anal. Appl., ,