ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES

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Weighted Partial Orderings Hanyu Li, Hu Yang and Hua Shao

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ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES

HANYU LI, HU YANG HUA SHAO

College of Mathematics and Physics Department of Mathematics & Physics

Chongqing University Chongqing University of Science and Technology Chongqing, 400030, P.R. China Chongqing, 401331, P.R. China

EMail:lihy.hy@gmail.com, yh@cqu.edu.cn EMail:shaohua.shh@gmail.com

Received: 21 January, 2008

Accepted: 19 May, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 15A45, 15A99

Key words: Weighted partial ordering, Matrix function, Singular value decomposition.

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, WeightedU DV^∗-decomposition and weighted spectral decomposition for rectangular matrices and their applications, Appl. Math. Comput. 198 (2008), pp. 150–162], some weighted partial orderings of matrices are compared with the orderings of their functions.

Acknowledgements: The authors would like to thank the editors and referees for their valuable com- ments and helpful suggestions, which improved the presentation of this paper.

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1. Introduction

LetC^m×ndenote the set ofm×ncomplex matrices,C^m×nr denote a subset ofC^m×n comprising matrices with rankr,C^m≥ denote a set of Hermitian positive semidefinite matrices of orderm, and C^m> denote a subset of C^m≥ consisting of positive definite matrices. Let Ir be the identity matrix of order r. GivenA ∈ C^m×n, the symbols A^∗, A^#_{M N}, R(A), and r(A) stand for the conjugate transpose, weighted conjugate transpose, range, and rank, respectively, ofA. Details for the concept of A^#_{M N} can be found in [11,13]. Moreover, unless otherwise specified, in this paper we always assume that the given weight matricesM ∈C^m×mandN ∈C^n×n.

In the following, we give some definitions of matrix partial orderings.

Definition 1.1. ForA, B ∈C^m×m, we say thatAis belowBwith respect to:

1. the Löwner partial ordering and writeA ≤_LB, wheneverB−A∈C^m≥. 2. the weighted Löwner partial ordering and writeA≤_{W L} B, wheneverM(B −

A)∈C^m≥.

Definition 1.2. ForA, B ∈C^m×n, we say thatAis belowB with respect to:

1. the star partial ordering and writeA≤^∗ B, wheneverA^∗A=A^∗BandAA^∗ = BA^∗.

2. the weighted star partial ordering and write A

#

≤ B, whenever A^#_{M N}A = A^#_{M N}B andAA^#_{M N} =BA^#_{M N}.

3. theW G-weighted star partial ordering and writeA

#

≤_{W G} B, wheneverM AB_{M N}^# ∈ C^m≥,N A^#_{M N}B ∈Cⁿ≥, andAA^#_{M N} ≤_{W L} AB_{M N}^# .

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4. theW GLpartial ordering and writeA≤_{W GL} B, whenever(AA^#_{M N})^1/2 ≤_{W L} (BB^#_{M N})^1/2andAB_{M N}^# = (AA^#_{M N})^1/2(BB_{M N}^# )^1/2.

5. the W GL2 partial ordering and write A ≤W GL2 B, whenever AA^#_{M N} ≤W L

BB_{M N}^# andAB_{M N}^# = (AA^#_{M N})^1/2(BB_{M N}^# )^1/2. 6. the minus partial ordering and write A

−

≤ B, whenever A⁻A = A⁻B and AA⁼ = BA⁼ for some (possibly distinct) generalized inversesA⁻, A⁼ of A (satisfyingAA⁻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, namelyA

−

≤B if and only ifr(B−A) = r(B)−r(A). 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 squaresA² andB² 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 matricesAandBrelate to the orderings of their generalized square A⁽²⁾ and B⁽²⁾, A⁽²⁾ = A(A^∗A)^1/2, in the sense of the GLpartial 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].

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In addition, Hauke and Markiewicz [5] also compared the star partial ordering A ≤^∗ B, G-star partial orderingA ≤^∗_G B, andGL partial orderingA ≤_GL B with the orderingsf(A)

∗

≤ f(B), f(A)

∗

≤_G f(B), and f(A) ≤_GL f(B), respectively.

Here,f(A)is a matrix function defined inA [7]. Legiša [8] also discussed the star partial ordering and surjective mappings onC^n×n. These results extended the work of Mathias [10] to some extent, who studied the relations between the Löwner partial orderingA≤LB and the orderingf(A)≤Lf(B).

In the present paper, based on the definition A⁽²⁾ = A(A^#_{M N}A)^1/2 (also called the generalized square ofA), we study how the partial orderings of two rectangular matricesAandBrelate to the orderings of their generalized squaresA⁽²⁾andB⁽²⁾in the sense of theW GLpartial 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, andA ≤_{W GL} B with the orderings f(A) ≤^# f(B), f(A) ≤^#_{W G} f(B), andf(A) ≤_{W GL} f(B), respectively. These works generalize the results of Hauke and Markiewicz [3,5].

Now we introduce the (M, N) weighted singular value decomposition [11, 12]

(MN-SVD) and the matrix functions based on the MN-SVD, which are useful in this paper,

Lemma 1.3. LetA ∈C^m×nr . Then there existU ∈C^m×m andV ∈C^n×nsatisfying U^∗M U =ImandV^∗N⁻¹V =Insuch that

(1.1) A=U

D 0 0 0

V^∗, where D = diag(σ₁, . . . , σ_r), σ_i = √

λ_i > 0, and λ₁ ≥ · · · ≥ λ_r > 0 are the nonzero eigenvalues of A^#_{M N}A = (N⁻¹A^∗M)A. Here, σ1 ≥ · · · ≥ σr > 0 are

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called the nonzero (M, N) weighted singular values of A. If, in addition, we let U = (U₁, U₂)andV = (V₁, V₂), whereU₁ ∈C^m×r andV₁ ∈C^n×r, then

(1.2) U₁^∗M U₁ =V₁^∗N⁻¹V₁ =I_r, A=U₁DV₁^∗.

Considering the MN-SVD, from [14], we can rewrite the matrix functionf(A) : C^m×n→C^m×nby way off(A) = U₁f(D)V₁^∗using the real functionf, wheref(D) is the diagonal matrix with diagonal elementsf(σ₁), . . . , f(σ_r). More information on the matrix function can be found in [14].

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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) theW GLpartial ordering coincides with the weighted Löwner partial ordering, i.e., for matricesA, B ∈C^m×msatisfyingM A, M B ∈C^m≥,

A≤_{W GL} Bif 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 matricesA, B ∈C^mm satisfyingM A, M B ∈C^m≥,

A≤_{W GL2} B if and only if A² ≤_{W GL} B².

On the set of rectangular matrices, for the generalized square of A, i.e., A⁽²⁾ = A(A^#_{M N}A)^1/2, 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. LetA, B ∈C^m×n,r(A) =a, andr(B) =b. Then (2.1) holds.

Proof. It is easy to find that the first conditions in the definitions ofW GL2partial ordering forAandBandW GLpartial ordering forA⁽²⁾andB⁽²⁾are equivalent. To prove the equivalence of the second conditions, let us use the MN-SVD introduced in Lemma1.3.

LetA =U₁D_aV₁^∗ andB =U₂D_bV₂^∗ be the MN-SVDs ofAandB, whereU₁ ∈ C^m×a,U₂ ∈C^m×b,V₁ ∈C^n×a, andV₂ ∈C^n×bsatisfyingU₁^∗M U₁ =V₁^∗N⁻¹V₁ =I_a and U₂^∗M U₂ = V₂^∗N⁻¹V₂ = I_b, and D_a ∈ C^a>, D_b ∈ C^b> are diagonal matrices.

Then

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AB_{M N}^# =(AA^#_{M N})^1/2(BB_{M N}^# )^1/2

⇔U1DaV₁^∗N⁻¹V2DbU₂^∗M

= (U₁D_aV₁^∗N⁻¹V₁D_aU₁^∗M)^1/2(U₂D_bV₂^∗N⁻¹V₂D_bU₂^∗M)^1/2

⇔U1DaV₁^∗N⁻¹V2DbU₂^∗M =U1DaU₁^∗M U2DbU₂^∗M

⇔V₁^∗N⁻¹V₂ =U₁^∗M U₂. (2.2)

Note that

A⁽²⁾ =A(A^#_{M N}A)^1/2 =U₁D_aV₁^∗(N⁻¹V₁D_aU₁^∗M U₁D_aV₁^∗)^1/2 (2.3)

=U₁D_aV₁^∗N⁻¹V₁D_aV₁^∗ =U₁D_a²V₁^∗. Similarly,

(2.4) B⁽²⁾ =U₂D²_bV₂^∗.

Then

A⁽²⁾(B⁽²⁾)^#_{M N} =(A⁽²⁾(A⁽²⁾)^#_{M N})^1/2(B⁽²⁾(B⁽²⁾)^#_{M N})^1/2

⇔U₁D²_aV₁^∗N⁻¹V₂D_b²U₂^∗M

= (U₁D²_aV₁^∗N⁻¹V₁D²_aU₁^∗M)^1/2(U₂D_b²V₂^∗N⁻¹V₂D²_bU₂^∗M)^1/2

⇔U₁D²_aV₁^∗N⁻¹V₂D_b²U₂^∗M =U₁D²_aU₁^∗M U₂D²_bU₂^∗M

⇔V₁^∗N⁻¹V₂ =U₁^∗M U₂, which together with (2.2) gives

AB_{M N}^# = (AA^#_{M N})^1/2(BB_{M N}^# )^1/2

⇔A⁽²⁾(B⁽²⁾)^#_{M N} = (A⁽²⁾(A⁽²⁾)^#_{M N})^1/2(B⁽²⁾(B⁽²⁾)^#_{M N})^1/2. Therefore, the proof is completed.

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Before studying the relation between theW GLpartial orderings forAandBand that for their generalized squares, we first introduce a lemma from [1].

Lemma 2.2. LetA, B ∈C^m≥. Then (a) If A² ≤_L B², then A≤_LB.

(b) If AB=BA and A≤_L B, then A² ≤_LB². Theorem 2.3. LetA, B ∈C^m×n,r(A) =a,r(B) =b, and

(a) A≤_{W GL} B, (b) A⁽²⁾ ≤_{W GL} B⁽²⁾,

(c) (AB_{M N}^# )^#_{M M} =AB_{M N}^# .

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 andW GL partial orderings, it suffices to show that

(2.5) (A⁽²⁾(A⁽²⁾)^#_{M N})^1/2 ≤_{W L} (B⁽²⁾(B⁽²⁾)^#_{M N})^1/2

⇒(AA^#_{M N})^1/2 ≤W L (BB_{M N}^# )^1/2. From the proof of Theorem2.1and the definition of weighted Löwner partial ordering, we have

(A⁽²⁾(A⁽²⁾)^#_{M N})^1/2 ≤_{W L} (B⁽²⁾(B⁽²⁾)^#_{M N})^1/2 (2.6)

⇔U₁D_a²U₁^∗M ≤_{W L} U₂D_b²U₂^∗M

⇔M U1D_a²U₁^∗M ≤LM U2D_b²U₂^∗M

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⇔M^1/2U₁D_a²U₁^∗M^1/2 ≤_LM^1/2U₂D²_bU₂^∗M^1/2

⇔M^1/2U₁D_aU₁^∗M^1/2M^1/2U₁D_aU₁^∗M^1/2

≤_LM^1/2U₂D_bU₂^∗M^1/2M^1/2U₂D_bU₂^∗M^1/2. Applying Lemma2.2(a) to (2.6) leads to

M^1/2U₁D_aU₁^∗M^1/2 ≤_L M^1/2U₂D_bU₂^∗M^1/2 (2.7)

⇔M U₁D_aU₁^∗M ≤_LM U₂D_bU₂^∗M

⇔M(AA^#_{M N})^1/2 ≤_L M(BB_{M N}^# )^1/2

⇔(AA^#_{M N})^1/2 ≤_{W L} (BB_{M N}^# )^1/2. Then, by (2.6) and (2.7), we show that (2.5) holds.

(ii). (a)and(c)⇒(b).

Similarly, combining with Theorem2.1and the definitions ofW GL2andW GL partial orderings, we only need to prove that

(2.8) (AA^#_{M N})^1/2 ≤W L (BB_{M N}^# )^1/2

⇒(A⁽²⁾(A⁽²⁾)^#_{M N})^1/2 ≤W L (B⁽²⁾(B⁽²⁾)^#_{M N})^1/2.

From the proof of Theorem2.1and the definition of weighted Löwner partial orderings, we have

(AA^#_{M N})^1/2 ≤_{W L}(BB_{M N}^# )^1/2 (2.9)

⇔U₁D_aU₁^∗M ≤_{W L} U₂D_bU₂^∗M

⇔M U₁D_aU₁^∗M ≤_LM U₂D_bU₂^∗M

⇔M^1/2U₁D_aU₁^∗M^1/2 ≤_LM^1/2U₂D_bU₂^∗M^1/2.

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According to (c), we have

(2.10) U₂D_bV₂^∗N⁻¹V₁D_aU₁^∗M =U₁D_aV₁^∗N⁻¹V₂D_bU₂^∗M.

Thus, together with (2.10) and (2.2), we can obtain (2.11) U2DbU₂^∗M U1DaU₁^∗M =U1DaU₁^∗M U2DbU₂^∗M

⇔M^1/2U₁D_aU₁^∗M^1/2M^1/2U₂D_bU₂^∗M^1/2

=M^1/2U₂D_bU₂^∗M^1/2M^1/2U₁D_aU₁^∗M^1/2. Applying Lemma2.2(b) to (2.11) and (2.9), we have

(2.12) M^1/2U1DaU₁^∗M^1/2M^1/2U1DaU₁^∗M^1/2

≤_LM^1/2U₂D_bU₂^∗M^1/2M^1/2U₂D_bU₂^∗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. LetA, B ∈C^m×nandr(B) =b > r(A) =a≥1. ThenA ≤^# Bif and only if there exist matricesU ∈ C^m×m andV ∈ C^n×nsatisfyingU^∗M U =I_m and V^∗N⁻¹V =I_nsuch that

A=U

D_a 0 0 0

V^∗ =U₁D_aV₁^∗,

B =U





Da 0 0

0 D 0

0 0 0



V^∗ =U₂

D_a 0

0 D

V₂^∗,

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where U₁ ∈ C^m×a, V₁ ∈ C^n×a and U₂ ∈ C^m×b, V₂ ∈ C^n×b denote the first a and b columns of U,V, respectively, and satisfy U₁^∗M U₁ = V₁^∗N⁻¹V₁ = I_a and U₂^∗M U₂ =V₂^∗N⁻¹V₂ =I_b, andD_a ∈C^a>andD∈C^b−a> are diagonal matrices.

Similarly to Lemma2.4, we can take the following form to characterize theW G- weighted star partial ordering. A detailed proof is omitted.

Lemma 2.5. LetA, B ∈ C^m×nandr(B) =b > r(A) =a ≥1. ThenA

#

≤_{W G} B if and only if there exist matricesU ∈ C^m×mandV ∈ C^n×n satisfyingU^∗M U =I_m andV^∗N⁻¹V =I_nsuch that

A=U

D_a 0 0 0

V^∗ =U1DaV₁^∗,

B =U





D_a⁰ 0 0

0 D 0

0 0 0



V^∗ =U₂

Da⁰ 0

0 D

V₂^∗,

whereU₁ ∈ C^m×a,V₁ ∈ C^n×a andU₂ ∈ C^m×b,V₂ ∈ C^n×b denote the firstaandb columns ofU,V, respectively, and satisfyU₁^∗M U₁ =V₁^∗N⁻¹V₁ =I_aandU₂^∗M U₂ = V₂^∗N⁻¹V₂ = I_b, and D_a, D_a⁰ ∈ C^a> and D ∈ C^b−a> are diagonal matrices, and D_a⁰ −D_a ∈C^a≥.

From the simultaneous weighted singular value decomposition of matrices [9], Lemma2.4, and Lemma2.5, we can derive the following theorem.

Theorem 2.6. LetA, B ∈C^m×n. Then (a) A

#

≤B ⇔M AB_{M N}^# ∈C_≥^m, N A^#_{M N}B ∈C_≥ⁿ,andAA^#_{M N} = (AA^#_{M N})^1/2(BB_{M N}^# )^1/2. (b) A

#

≤_{W G} B ⇔ M AB_{M N}^# ∈ C_≥^m, N A^#_{M N}B ∈ C_≥ⁿ, and (AA^#_{M N})^1/2 ≤_{W L} (BB^#_{M N})^1/2.

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Considering Definition 1.2(4) 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 forW G-weighted star partial ordering using Lemma2.5.

Theorem 2.7. LetA, B ∈C^m×n,r(A) =a, andr(B) =b. Then A⁽²⁾

#

≤_{W G}B⁽²⁾if and only ifA

#

≤_{W G}B.

Proof. Let the MN-SVDs ofAandBbe as in the proof of Theorem2.1. Considering Lemma1.3, from (2.3), (2.4), and Lemma2.5, we have

A⁽²⁾ =U₁D_a²V₁^∗ =U

D²_a 0 0 0

V^∗, B⁽²⁾ =U₂D_b²V₂^∗ =U

D²_b 0 0 0

V^∗. In this case, the MN-SVDs ofAandB can be rewritten as

A=U

D_a 0 0 0

V^∗, B =U

D_b 0 0 0

V^∗. Thus, from Lemma2.5, we have

A⁽²⁾

#

≤_{W G} B⁽²⁾ ⇒A

#

≤_{W G}B.

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Conversely, from Lemma2.5,A≤^#_{W G} Bis equivalent to A=U

Da 0 0 0

V^∗, B =U





D_a⁰ 0 0

0 D 0

0 0 0



V^∗. Then

A⁽²⁾=U

D_a² 0 0 0

V^∗, B⁽²⁾ =U





D_b² 0 0 0 D² 0

0 0 0



V^∗. Therefore, from Lemma2.5again, the proof is completed.

The characterization of the weighted star partial ordering can be obtained similarly using Lemma2.4, and is given in the following theorem.

Theorem 2.8. LetA, B ∈C^m×n,r(A) =a, andr(B) =b. Then A⁽²⁾

#

≤B⁽²⁾if and only ifA

#

≤B.

The following result was presented by Liu [9]. It is useful for studying the relation between the minus ordering forAandB and that forA⁽²⁾ andB⁽²⁾.

Lemma 2.9. LetA, B ∈C^m×n. Then

A≤^#B if and only ifA≤⁻B,

(AB_{M N}^# )^#_{M M} =AB^#_{M N}, and(A^#_{M N}B)^#_{N N} =A^#_{M N}B.

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Theorem 2.10. LetA, B ∈ C^m×n, r(A) = a, r(B) = b, (AB_{M N}^# )^#_{M M} = AB_{M N}^# , and(A^#_{M N}B)^#_{N N} =A^#_{M N}B. Then

A⁽²⁾

−

≤B⁽²⁾if and only ifA

−

≤B.

Proof. According to(AB_{M N}^# )^#_{M M} = AB^#_{M N},(A^#_{M N}B)^#_{N N} = A^#_{M N}B, the proof of Theorem 5.3.2 of [9], and the simultaneous unitary equivalence theorem [7], we have

A=U

E_c 0 0 0

V^∗, B =U

F_c 0 0 0

V^∗,

where U ∈ C^m×mand V ∈ C^n×n satisfy U^∗M U = Im and V^∗N⁻¹V = In, and E_c ∈C^c×c≥ andF_c are real diagonal matrices,c= max{a, b}.

As in (2.3) and (2.4), we can obtain A⁽²⁾ =U

E_c² 0

0 0

V^∗, B⁽²⁾ =U

F_c|F_c| 0

0 0

V^∗. Thus, it is easy to verify that

(A⁽²⁾(B⁽²⁾)^#_{M N})^#_{M M} =A⁽²⁾(B⁽²⁾)^#_{M N} and ((A⁽²⁾)^#_{M N}B⁽²⁾)^#_{N N} = (A⁽²⁾)^#_{M N}B⁽²⁾. As a result,

A⁽²⁾≤⁻B⁽²⁾ ⇔A⁽²⁾≤^#B⁽²⁾. By Theorem2.8and Lemma2.9, the proof is completed.

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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 whichr[f(A)] = r(A), i.e., functions for which f(x) = 0 only for x= 0. These functions are said to be nondegenerating.

The following properties off gathered in Lemma3.1will 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) R(A) =R(f(A)).

(b) AB_{M N}^# = (AA^#_{M N})^1/2(BB_{M N}^# )^1/2

⇔f(A)f(B_{M N}^# ) = f((AA^#_{M N})^1/2)f((BB_{M N}^# )^1/2).

Proof. (a). From the MN-SVD ofA, i.e., (1.2), and the property off, we have R(A) =R(U₁DV₁^∗) =R(U₁) =R(U₁f(D)V₁^∗) = R(f(A)).

(b). Similar to the proof of Theorem2.1, letA =U1DaV₁^∗andB =U2DbV₂^∗be the MN-SVDs ofAandB respectively. Considering the definition of matrix functions, we obtain

f(A)f(B_{M N}^# ) =f((AA^#_{M N})^1/2)f((BB_{M N}^# )^1/2)

⇔U₁f(D_a)V₁^∗N⁻¹V₂f(D_b)U₂^∗M =U₁f(D_a)U₁^∗M U₂f(D_b)U₂^∗M

⇔V₁^∗N⁻¹V₂ =U₁^∗M U₂, which together with (2.2) implies the proof.

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In the following theorems, we compare some weighted partial orderings of matrices with orderings of their functions.

Theorem 3.2. LetA, B ∈C^m×nand letf be a positive one-to-one function. Then A≤^# B if and only if f(A)≤^# f(B).

Proof. From Definition1.2(2) and Lemma2.4, we have thatA

#

≤B is equivalent to AB_{M N}^# =U₁D_a²U₁^∗M =AA^#_{M N} and A^#_{M N}B =N⁻¹V₁D_a²V₁^∗ =A^#_{M N}A, andf(A)

#

≤f(B)is equivalent to

f(A)f(B)^#_{M N} =U1f(Da)²U₁^∗M =f(A)f(A^#_{M N}) and f(A)^#_{M N}f(B) = N⁻¹V1f(Da)²V₁^∗ =f(A^#_{M N})f(A).

Then, using the properties off, the proof is completed.

Theorem 3.3. LetA, B ∈C^m×nand letf be a positive strictly increasing function.

Then

A

#

≤_{W G} B if and only if f(A)

#

≤_{W G} f(B).

Proof. From Definition 1.1(2), Definition 1.2(3), and Lemma 2.5, we obtain that A

#

≤_{W G}B is equivalent to

M AA^#_{M N} =M U1D²_aU₁^∗M ≤LM U1DaDa⁰U₁^∗M =M AB_{M N}^# , M AB^#_{M N} =M U1DaDa⁰U₁^∗M ∈C^m≥,

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and

N A^#_{M N}B =V₁D_aD_a⁰V₁^∗ ∈Cⁿ≥; andf(A)

#

≤_{W G} f(B)is equivalent to

M f(A)f(A)^#_{M N} =M U1f(Da)²U₁^∗M ≤L M U1f(Da)f(Da⁰)U₁^∗M

=M f(A)f(B)^#_{M N},

M f(A)f(B)^#_{M N} =M U₁f(D_a)f(D_a⁰)U₁^∗M ∈C^m≥

and

N f(A)^#_{M N}B =V₁f(D_a)f(D_a⁰)V₁^∗ ∈Cⁿ≥. Therefore, the proof follows from the property off.

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 GLpartial ordering of their functions to a suitable problem involving the weighted Löwner partial ordering. Thus, from Definition1.1(2), Definition1.2(4), and Lemma 3.1, we can deduce the following theorem.

Theorem 3.4. LetA, B ∈C^m×nand letf be a positive strictly increasing function.

The following statements are equivalent:

(a) A≤_{W GL} B if and only if f(A)≤_{W GL} f(B).

(b) (AA^#_{M N})^1/2 ≤_{W L} (BB_{M N}^# )^1/2if and only iff((AA^#_{M N})^1/2)≤_{W L} f((AA^#_{M N})^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 Section2. For example, if f(t) = t², then f(A) = U1D²V₁^∗ = A⁽²⁾, hence, in this case, Theorem 3.2 and Theorem 3.3 will reduce to Theorem2.8and Theorem2.7, respectively.

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References

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