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Nonnegative splittings for rectangular matrices
Computers and Mathematics with Applications 67 (2014) 136–144
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Nonnegative splittings for rectangular matrices Debasisha Mishra Department of Mathematics, National Institute of Technology Raipur, Raipur- 492010, India
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Article history: Received 11 March 2013 Received in revised form 17 September 2013 Accepted 26 October 2013 Keywords: Moore–Penrose inverse Linear system Nonnegative splitting Proper splitting Convergence theorem Comparison theorem
abstract The extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first. Different convergence and comparison theorems for the proper nonnegative splittings are established. The notion of double nonnegative splitting is then generalized to rectangular matrices. Finally, different convergence and comparison results are presented for this decomposition. The case for singular square matrices is also studied. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The theory of splittings for square nonsingular matrices and its relationship with the solution of system of linear equations is quite well-known. Standard iterative methods like the Jacobi, Gauss–Seidel and successive over-relaxation methods for solving a square nonsingular system of linear equations Ax = b, arise from different choices of real square matrices U and V , where A = U − V and b is a real n-vector. The book by Varga [1] contains several splittings such as regular and weak regular splittings. A decomposition A = U − V of a real square nonsingular matrix A is (i) regular splitting if U −1 exists, U −1 ≥ O and V ≥ O [1], (ii) weak regular splitting if U −1 exists, U −1 ≥ O and U −1 V ≥ O [2,1], (iii) nonnegative if U −1 exists and U −1 V ≥ O [3], where the comparison is entrywise and O is the null matrix. The theory of nonnegative splittings is analyzed in [4,3,5–7]. Also, Csordas and Varga [8], Elsner [9], Song [3,5], Song and Song [6], Woźnicki [10] and many others have proved various comparison results for different matrix splittings. Berman and Plemmons [11] then extended the concept of splittings to rectangular matrices and called it as a proper splitting. A decomposition A = U − V of A ∈ Rm×n is called a proper splitting [11] if R(A) = R(U ) and N (A) = N (U ), where R(A) and N (A) denote the range space and the null space of A. Linear systems of the form Ax = b,
(1.1)
where A is a real square singular or real sparse or real rectangular matrix appear in many areas of mathematics. For example rectangular/singular systems arise by applying finite difference methods to partial differential equations such as the Neumann Problem and Poisson’s equation. The iteration x(i+1) = U Ď Vx(i) + U Ď b, E-mail address: [email protected]. 0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.10.012
(1.2)
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is often employed to solve these systems, where BĎ means the Moore–Penrose inverse of B (see next section for the definition). The scheme (1.2) is said to be convergent if the spectral radius of U Ď V is less than 1, and U Ď V is called the iteration matrix. The authors of [11] showed that if A = U − V is a proper splitting, then the scheme (1.2) converges to AĎ b, the least square solution of minimum norm for any initial vector x0 if and only if the spectral radius of U Ď V is less than 1. (See Corollary 1, [11].) However, it is not true for any initial vector x0 . As if x0 = 0 or x0 ∈ N (V ), then the iterative sequence will not move further. Hence the initial vector x0 should not be a zero-vector and should not lie in the null space of V . (Interested readers can have a look at the introductory part of the article [11] for the reason and importance of choosing proper splittings.) We also remark that the scheme Y (j+1) = U Ď VY (j) + U Ď also converges to AĎ under analogous conditions for a suitable initial matrix Y 0 . Very recently, the authors of [12] extended the notion of regular and weak regular splittings to rectangular matrices and the respective definitions are recalled next. A decomposition A = U − V of A ∈ Rm×n is called a proper regular splitting if it is a proper splitting such that U Ď ≥ O and V ≥ O. Similarly, A = U − V is called proper weak regular splitting if it is a proper splitting such that U Ď ≥ 0 and U Ď V ≥ O. Note that Berman and Plemmons [11] proved a convergence theorem for these splittings without specifying the types of matrix decompositions. Using the notion of proper regular splitting, Theorem 3, [11] can be now rewritten as follows. Let A = U − V be a proper regular splitting of A ∈ Rm×n . Then AĎ ≥ O if and only if ρ(U Ď V ) < 1. In [12], one can find comparison results for these splittings and their applications to the double splitting theory. But, in this paper, our plan is to introduce another new decomposition which is an extension of so-called nonnegative splitting for square nonsingular matrices, and is more general than the proper regular and proper weak regular splittings (see Section 3 for more discussion). Eq. (1.1) can also be solved using double decomposition of A. (A decomposition of a real m × n matrix of the form A = P −R−S is called double decomposition; for the real square nonsingular case it is called double splitting if P is nonsingular.) The idea of double splitting was first introduced by Woźnicki [13] for square nonsingular matrices. This notion was then extended by Jena et al. [12] for real m × n matrices. Shen et al. [14], Miao and Zheng [15] and Song and Song [6] have studied convergence and comparison theorems of real square nonsingular matrices using double splittings. In particular, Song and Song [6] proved that the double splitting is convergent if and only if the single splitting is convergent for the nonnegative splittings. In this paper, we are going to extend the same result for real rectangular matrices along with a comparison result. The central idea of this paper is to study the theory of nonnegative splittings for rectangular matrices. The organization is as follows. In Section 2, we list all relevant definitions, notation and some earlier results which we use in the paper. The main results are given in Sections 3 and 4. Section 3 introduces the generalization of nonnegative splitting to rectangular matrices, and then discusses convergence and comparison theorems for these decompositions. In Section 4, we propose the notion of double proper nonnegative splittings for real m × n matrices. Then convergence results for double proper nonnegative splittings are established. At last, we obtain a comparison theorem for two different linear systems. Section 5 discusses the group inverse analogue of a few main results mentioned in Sections 3 and 4 for square singular matrices. Finally, we end up with conclusions. 2. Preliminaries Let Rn denote the n dimensional real Euclidean space and Rn+ denote the nonnegative orthant in Rn . For a real m × n matrix A, i.e., A ∈ Rm×n , the matrix G satisfying the four equations known as Penrose equations: AGA = A, GAG = G, (AG)T = AG and (GA)T = GA is called the Moore–Penrose inverse of A (BT denotes the transpose of B). It always exists and is unique, and is denoted by AĎ . A ∈ Rm×n is said to be semi-monotone if AĎ ≥ O. The group inverse of a matrix A ∈ Rn×n (if it exists), denoted by A# is the unique matrix X satisfying A = AXA, X = XAX and AX = XA. Equivalently, A# is the unique matrix X which satisfies XAx = x for all x ∈ R(A) and Xy = 0 for all y ∈ N (A). The index of a real square matrix A is the least nonnegative integer k such that rank(Ak+1 ) = rank(Ak ). It is well known that A# exists if and only if index of A is 1 (i.e., R(A) ⊕ N (A) = Rn ). Let A ∈ Rn×n be of index k. Then, the Drazin inverse of A is the unique matrix AD ∈ Rn×n which satisfies the equations Ak+1 AD = Ak , AD AAD = AD and AAD = AD A. A ∈ Rn×n is said to be group monotone if A# exists and A# ≥ O. Similarly, it is called Drazin monotone if AD ≥ O. When A is a square nonsingular, then AĎ = A−1 = A# = AD , and a semi-monotone matrix (as well as a group monotone matrix or a Drazin monotone matrix) becomes a monotone matrix (i.e., A−1 exists and A−1 ≥ O). (See the book by Berman and Plemmons, [2] for more details on monotone matrices and their generalizations.) For A, B, C ∈ Rm×n , we say A is nonnegative if A ≥ O, and B ≥ C if B − C ≥ O. We denote a nonnegative vector x as x ≥ 0. Let K , L be complementary subspaces of Rp , i.e., K ⊕ L = Rp . Then PK ,L denotes the (not necessarily orthogonal) projection of Rp onto K along L. Thus PK2,L = PK ,L , R(PK ,L ) = K and N (PK ,L ) = L. If in addition, K ⊥ L, PK ,L will be denoted by PK . A few properties of AĎ and A# [16] are listed here: R(AT ) = R(AĎ ); N (AT ) = N (AĎ ); AAĎ = PR(A) ; AĎ A = PR(AT ) ; R(A) = R(A# ); N (A) = N (A# ); AA# = PR(A),N (A) . In particular, if x ∈ R(A∗ ) then x = AĎ Ax and if x ∈ R(A) then x = A# Ax. The spectral radius of A ∈ Rn×n , denoted by ρ(A) is defined by ρ(A) = max1≤i≤n |λi |, where λ1 , λ2 , . . . , λn are the eigenvalues of A. The following results will be helpful to prove our main result. The first one is a part of Theorem 1, [11] and Theorem 3.1, [17] which expresses A and AĎ in terms of U and V using the proper splitting A = U − V . Theorem 2.1. Let A = U − V be a proper splitting. Then (a) AAĎ = UU Ď ; AĎ A = U Ď U.
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(b) A = U (I − U Ď V ) = (I − VU Ď )U. (c) I − U Ď V and I − VU Ď are invertible. (d) AĎ = (I − U Ď V )−1 U Ď = U Ď (I − VU Ď )−1 . The next lemma shows a relation between the eigenvalues of U Ď V and AĎ V . Lemma 2.2 (Lemma 2.6, [18]). Let A = U − V be a proper splitting of A ∈ Rm×n . Let µi , 1 ≤ i ≤ s and λj , 1 ≤ j ≤ s be the eigenvalues of the matrices U Ď V (VU Ď ) and AĎ V (VAĎ ) respectively. Then for every j, we have 1 + λj ̸= 0. Also, for every i, there λ
µ
j i and for every j, there exists i such that λj = 1−µ . exists j such that µi = 1+λ j i
The next theorem is a part of the Perron–Frobenius theorem. Theorem 2.3 (Theorem 2.20, [1]). Let A ≥ O. Then A has a nonnegative real eigenvalue equal to its spectral radius. Another result which relates spectral radius of two nonnegative matrices is given below. Theorem 2.4 (Theorem 2.21, [1]). Let A ≥ B ≥ O. Then ρ(A) ≥ ρ(B). In the theory of nonnegative matrices, the following three results are well known. Theorem 2.5 (Theorem 3.16, [1]). Let X ∈ Rn×n and X ≥ O. Then ρ(X ) < 1 if and only if (I − X )−1 exists and (I − X )−1 = ∞ k k=0 X ≥ O. Lemma 2.6 (Corollary 3.2, [4]). Let B ≥ O and x ≥ 0 be such that Bx − α x ≥ 0. Then α ≤ ρ(B). Lemma 2.7 (Lemma 2.2, [14]). Let X =
B I
C O
≥ O and ρ(B + C ) < 1. Then ρ(X ) < 1.
3. Proper nonnegative splittings Let us begin this section with the definition of a Z -matrix and an M-matrix. A square matrix whose off-diagonal elements are non-positive is called a Z -matrix. A Z -matrix A is called a nonsingular M-matrix if A is monotone. Equivalently, a Z -matrix A is called a nonsingular M-matrix if A = sI − B, where s > ρ(B) and B ≥ O. More on M-matrices can be found in Chapter 6 of [2]. Setting U = sI and V = B, we always have a nonnegative splitting. The problem is when s = ρ(B) and in that case A becomes a singular M-matrix. The Laplacian matrix of an undirected graph is a singular M-matrix. To deal with this situation and also in more general way i.e., the case when A is a real rectangular matrix, we propose the following definition. Definition 3.1. A decomposition A = U − V of A ∈ Rm×n is called a proper nonnegative splitting if it is a proper splitting such that U Ď V ≥ O. The class of matrices having a positive real number in all entries always have proper nonnegative splittings. We also present another two classes of matrices which always have the same decomposition. Example 3.2. Let A ∈ Rn×n and be of the form (n−r )×r
(n−r )×(n−r )
B O2
O1 O3
where B is a nonsingular M-matrix of order r , O1 ∈ Rr ×(n−r ) , O2 ∈
R and O3 ∈ R are zero matrices. Then A is a singular Z -matrix. Take U = diag(b11 , b22 , . . . , brr , 0, 0, . . . , 0) where B = (bij ) and V = U − A. Then A = U − V is a proper nonnegative splitting. Example 3.3. Let A ∈ R2n×n be partitioned as A = U =
D1 D2
B C
where B and C are real n × n nonsingular M-matrices. Setting
, where D1 = diag(b11 , b22 , . . . , bnn ) and D2 = diag(c11 , c22 , . . . , cnn ) with B = (bij ) and C = (cij ), respectively.
Then V = U − A and thus we have A = U − V is a proper nonnegative splitting. We now present a convergence result for the proper nonnegative splitting which is an extension of Lemma 2.1, [3] for real rectangular matrices. Note that the converse of the result given below is not true in general. Nevertheless, this is true by assumption of one more condition AĎ A ≥ O. ρ(AĎ U )−1
Lemma 3.4. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n and AĎ U ≥ O. Then ρ(U Ď V ) = ρ(AĎ U ) < 1. Proof. Since A = U − V is a proper nonnegative splitting, so A = U − V is a proper splitting with U Ď V ≥ O. By Theorem 2.3, there exists a nonnegative vector x (x ̸= 0) such that U Ď Vx = ρ(U Ď V )x. Hence x ∈ R(U Ď ) = R(U T ) = R(AT ) so that U Ď Ux = x. By Theorem 2.1(d), we also have AĎ = (I − U Ď V )−1 U Ď . So AĎ U = (I − U Ď V )−1 U Ď U. Then
(I − U Ď V )−1 U Ď Ux = (I − U Ď V )−1 x =
1 1 − ρ(U Ď V )
x = AĎ Ux. ρ(AĎ U )−1
This implies 1−ρ(1U Ď V ) ≥ 0 and is an eigenvalue of AĎ U. Hence 0 ≤ 1−ρ(1U Ď V ) ≤ ρ(AĎ U ) i.e., ρ(U Ď V ) ≤ ρ(AĎ U ) . Again, the condition AĎ U ≥ O implies existence of a nonnegative vector y (y ̸= 0) such that AĎ Uy = ρ(AĎ U )y. Then y ∈ R(AT ) = R(U T ).
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
Therefore (I − U Ď V )−1 U Ď Uy = (I − U Ď V )−1 y = ρ(AĎ U )y. So, we have ρ(AĎ1U )y = y − U Ď Vy i.e., U Ď Vy = Ď
ρ(U V ) ≥
ρ(AĎ U )−1 . ρ(AĎ U )
Ď
Hence ρ(U V ) =
ρ(AĎ U )−1 ρ(AĎ U )
< 1.
139
ρ(AĎ U )−1 ρ(AĎ U )
y. Thus
Ď j Ď Since ∀ x ∈ Rn+ ∩ R(AT ), AĎ Ux = (I − U Ď V )−1 U Ď Ux = (I − U Ď V )−1 x = j=0 (U V ) x ≥ 0 (if ρ(U V ) < 1). Hence the n T converse of the above theorem is also true on R+ ∩ R(A ). Another convergence result for the proper nonnegative splitting is obtained below. A similar proof as in the above theorem can be provided. However, we present a new proof to this.
∞
ρ(AĎ V )
Lemma 3.5. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n . Then AĎ V ≥ O if and only if ρ(U Ď V ) = 1+ρ(AĎ V ) < 1. Proof. We have AĎ V ≥ O and U Ď V ≥ O. Let λ and µ be any nonnegative eigenvalues of AĎ V and U Ď V , respectively. Let λ λ f (λ) = 1+λ , λ ≥ 0. Then f is a strictly increasing function. Then by Lemma 2.2, µ = 1+λ . So, µ attains its maximum when
λ is maximum. But λ is maximum when λ = ρ(AĎ V ). As a result, the maximum value of µ is ρ(U Ď V ). Hence, ρ(U Ď V ) = ρ(AĎ V ) < 1. 1+ρ(AĎ V ) Conversely, A = U − V is splitting yields AĎ =(I − U Ď V )−1 U Ď . By Theorem 2.5, the condition ρ(U Ď V ) < 1 a∞proper ∞ Ď k Ď k ( U V ) . Therefore AĎ V = implies that (I − U Ď V )−1 = k=1 (U V ) ≥ O. k=0 The above lemma admits the following corollary in case of nonsingular matrices. Corollary 3.6 (Lemma 2.2, [3]). Let A = U − V be a nonnegative splitting of A ∈ Rn×n . Then A−1 V ≥ O if and only if ρ(U −1 V ) = ρ(A−1 V ) 1+ρ(A−1 V )
< 1.
We then present certain properties of proper nonnegative splitting. Theorem 3.7. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n and AĎ V ≥ O. Then (a) (I − U Ď V )−1 ≥ O. (b) (I − U Ď V )−1 ≥ I. (c) AĎ V ≥ U Ď V . Proof. (a) By Lemma 3.5, we have ρ(U Ď V ) < 1. Then, Theorem 2.5 implies that (I − U Ď V )−1 exists and (I − U Ď V )−1 = ∞ Ď k condition U Ď V ≥ O implies (I − U Ď V )−1 ≥ O. k=0 (U V ) . Finally the ∞ ∞ Ď k Ď k (b) (I − U Ď V )−1 = k=0 (U V ) = I + k=1 (U V ) . As the decomposition is proper nonnegative, so the second term in the above expression is nonnegative. Hence the claim follows. (c) From Theorem 2.1(d), we have AĎ = (I − U Ď V )−1 U Ď . So (I − U Ď V )AĎ = U Ď . Then post-multiplying by V , we have AĎ V − U Ď V = U Ď VAĎ V . Since AĎ V and U Ď V are nonnegative, so AĎ V − U Ď V ≥ O. Hence AĎ V ≥ U Ď V . The next theorem generalizes some parts of Lemma 2.6, [5] to rectangular matrices. Theorem 3.8. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n . Then the following are equivalent:
ρ(U Ď V ) < 1; AĎ U − AĎ A ≥ O; AĎ U ≥ AĎ A; AĎ V ≥ U Ď V ; AĎ V ≥ O; ρ(AĎ V ) (f) ρ(U Ď V ) = 1+ρ(AĎ V ) .
(a) (b) (c) (d) (e)
Proof. ⇒ (b) A = U − V is a proper splitting implies AĎ = (I − U Ď V )−1 U Ď . Again ρ(U Ď V ) < 1 yields that (I − U Ď V )−1 = ∞ (a) Ď T T Ď T T Ď ( U V )k . Since k=0 ∞ N (A) = N (U ) so N (A) ⊆ N (V ) i.e. R(V ) ⊆ R(A ). Hence U UV = V which gives VU U = V . Therefore AĎ U = U Ď U + k=1 (U Ď V )k ≥ O. The fact U Ď V ≥ O implies that AĎ U − AĎ A ≥ O. (b) ⇒ (c) Clearly. (c) ⇒ (d) Since U Ď V ≥ O, post-multiplying U Ď V to AĎ U ≥ AĎ A we get AĎ UU Ď V ≥ AĎ AU Ď V = U Ď UU Ď V . Also, the condition R(A) = R(U ) implies R(V ) ⊆ R(A) = R(U ). Hence UU Ď V = V . We thus have AĎ V ≥ U Ď V . (d) ⇒ (e) Since U Ď V ≥ O, so AĎ V ≥ U Ď V ≥ O. (e) ⇒ (f) From Lemma 3.5. (f) ⇒ (a) Obviously. Another result for a convergent proper nonnegative splitting is shown next. Theorem 3.9. Let A = U − V be a convergent proper nonnegative splitting of A ∈ Rm×n . If AĎ ≥ O, then (a) AĎ ≥ U Ď ; (b) AĎ V ≥ U Ď V ;
ρ(AĎ V )
(c) ρ(U Ď V ) = 1+ρ(AĎ V ) .
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D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
The proof of (c) is analogous to the proof of sufficient part of Lemma 3.5, while (a) and (b) follows using the properties of a proper splittings. The condition AĎ V2 ≥ AĎ V1 ≥ O gives ρ(AĎ V2 ) ≥ ρ(AĎ V1 ) by Theorem 2.4. Again, using the same idea as in the proof of Lemma 3.5, we have the following comparison theorem for two proper nonnegative splittings. Theorem 3.10. Let A = U1 − V1 = U2 − V2 be two proper nonnegative splittings of A ∈ Rm×n . If AĎ V2 ≥ AĎ V1 ≥ O, then Ď
Ď
1 > ρ(U2 V2 ) ≥ ρ(U1 V1 ). The next result is the final result of this section, and is a generalization of Theorem 2.17, [5]. The proof is analogous to nonsingular case and thus omitted. Theorem 3.11. Let A = U1 − V1 = U2 − V2 be two convergent proper nonnegative splittings of A ∈ Rm×n . Let x ≥ 0 and z ≥ 0 be Ď Ď Ď Ď Ď such that U1 V1 x = ρ(U1 V1 )x and U2 V2 z = ρ(U2 V2 )z. Suppose that AĎ ≥ O and either V1 x ≥ 0 with ρ(U1 V1 )x > 0 or V2 z ≥ 0 Ď
Ď
Ď
with z > 0 and ρ(U2 V2 )z > 0. Further, assume that there exists α, 0 < α ≤ 1 such that U2 ≤ α U1 . Then
ρ(U1Ď V1 ) ≤ ρ(U2Ď V2 ) < 1,
when α = 1 and
ρ(U1Ď V1 )
when 0 < α < 1.
ρ(U2Ď V2 )
<
< 1,
A decomposition A = U − V of A ∈ Rm×n is called proper nonnegative splitting of 2nd type if it is a proper splitting with VU Ď ≥ O. In case of nonsingular matrices, this is called as nonnegative splitting of 2nd type. We also mention that proper nonnegative splitting of 1st type is same as proper nonnegative splitting. In the literature, these types of splittings have different names. However, we will use these names. Now, we conclude this section with the remark that similar results for the proper nonnegative splitting of 2nd type can also be obtained with some minor changes. 4. Double proper nonnegative splittings Analogous to the idea of Jena et al. [12] the double decomposition A = P − R − S of A ∈ Rm×n to (1.1) leads to the following iterative scheme spanned in three iterates: xi+1 = P Ď Rxi + P Ď Sxi−1 + P Ď b,
i > 0.
(4.1)
Then
xi+1 xi
where yi+1 =
=
P ĎR I
i+1
x
i
x
P ĎS O
, yi =
xi
i −1
x xi
i−1
x
+
,W =
P Ďb , O
PĎR I
PĎS O
and d =
PĎb O
i.e.,
yi+1 = Wyi + d.
(4.2)
Then the above scheme is convergent if ρ(W ) < 1. In this case, we say A = P − R − S is a convergent double decomposition. Jena et al. [12] introduced two double decompositions called double proper regular splitting and double proper weak regular splitting. A double decomposition A = P − R + S is called double proper regular splitting if R(A) = R(P ), N (A) = N (P ), P Ď ≥ O, R ≥ O and S ≤ O. A double decomposition A = P − R + S is called double proper weak regular splitting if R(A) = R(P ), N (A) = N (P ), P Ď ≥ 0, P Ď R ≥ O and P Ď S ≤ O. Note that Jena et al. considered the decomposition A = P − R + S while we are considering A = P − R − S. The definition of a new double decomposition called double proper nonnegative splitting is now proposed. Definition 4.1. A double decomposition A = P − R − S of A ∈ Rm×n is called double proper nonnegative splitting if R(A) = R(P ), N (A) = N (P ), P Ď R ≥ O and P Ď S ≥ O. In case of nonsingular matrices, we have the following definition. Definition 4.2 (Definition 1.3, [6]). A double decomposition A = P − R − S of A ∈ Rn×n is called double nonnegative splitting if P −1 R ≥ O and P −1 S ≥ O. Setting P = U and R − S = V in Definition 4.1, we obtain proper nonnegative splittings. We first prove a convergence result which relates convergence of single and double decompositions. Theorem 4.3. Let A = P − R − S be a double proper nonnegative splitting of A ∈ Rm×n . Then ρ(W ) < 1 if and only if ρ(U Ď V ) < 1, where U = P and V = R + S. Proof. Since A = P − R − S be a double proper nonnegative splitting, so W ≥ O. Then, Theorem 2.5 says that ρ(W ) < 1 if and only if (I − W )−1 exists and (I − W )−1 ≥ O. A = P − (R + S ) = U − V is a proper splitting implies that [I − P Ď (R + S )]−1
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
141
exists (by Theorem 2.1(c)). By block matrix computation, we get
(I − W )
−1
=
[I − P Ď (R + S )]−1 O
=
Ď
[I − P (R + S )]−1 [I − P Ď (R + S )]−1
O [I − P Ď (R + S )]−1
I I
P ĎS I − P ĎR
[I − P Ď (R + S )]−1 P Ď S . [I − P Ď (R + S )]−1 (I − P Ď R)
If (I − W )−1 ≥ O, then [I − P Ď (R + S )]−1 ≥ O. By Theorem 2.5, we get ρ(U Ď V ) = ρ(P Ď (R + S )) < 1. Ď Ď −1 −1 Conversely, if ρ(U Ď V ) = ρ(P (R1 + S )) < 1, then [I − P (R + S )] ≥ O. We will now show that (I − W ) exists. Let x I − PĎR −P Ď S 1 2 T Ď 1 Ď 2 1 2 (I − W )x = 0. So −I = 0, where x = (x , x ) . Then, we have I − P Rx = P Sx and x = x . For simplicity 2 I x
put x1 = x2 = y. So y = (P Ď R + P Ď S )y = P Ď (R + S )y = U Ď Vy. Hence y ∈ R(U T ) = R(AT ). Again, the condition y = U Ď Vy implies Uy = UU Ď Vy = Vy since we have R(V ) ⊆ R(A) (which follows from the condition R(A) = R(U )). Thus Uy = Vy which yields (U − V )y = Ay = 0. Therefore we get x = 0. Hence I − W is invertible. ∞ y ∈ĎN (A) which in∞turnĎ impliesj yĎ= 0 and then j Ď Also [I − P Ď (R + S )]−1 (I − P Ď R) = ( P ( R + S )) − ( P ( R + S )) P R = I + P ( R + S )− P Ď R +[P Ď (R + S )]2 − P Ď (R + S )P Ď R + j=0 j=0
· · · = I + P Ď S + P Ď (R + S )(P Ď S ) + [P Ď (R + S )]2 (P Ď S ) + · · · ≥ I ≥ O since P Ď R and P Ď S are nonnegative. The condition [I − P Ď (R + S )]−1 (I − P Ď R) ≥ O implies (I − W )−1 ≥ O. Hence ρ(W ) < 1. As a corollary to the above result Theorem 2.2, [6] is obtained and is shown next. Corollary 4.4 (Theorem 2.2, [6]). Let A = P − R − S be a double nonnegative splitting of A ∈ Rn×n . Then ρ(W ) < 1 if and only if ρ(U −1 V ) < 1, where U = P and V = R + S. We then present a convergence theorem for the double proper nonnegative splittings. Thereafter a similar result is obtained for the double nonnegative splitting as a corollary. Theorem 4.5. Let AĎ P ≥ O. If A = P − R − S is a double proper nonnegative splitting of A ∈ Rm×n , then ρ(W ) < 1. Proof. A = P − R − S is a double proper nonnegative splitting implies W =
PĎR I
PĎS O Ď
≥ O. Setting U = P and V = R + S,
we get A = U − V is a proper nonnegative splitting. Then by Lemma 3.4, we have ρ(P (R + S )) = ρ(U Ď V ) < 1. By Lemma 2.7, it now follows that ρ(W ) < 1. Corollary 4.6. Let A−1 P ≥ O. If A = P − R − S is a double nonnegative splitting of A ∈ Rn×n , then ρ(W ) < 1. Theorem 4.5 is also true if the condition AĎ P ≥ O is replaced by AĎ (R + S ) ≥ O. (Then Lemma 3.5 will be used in the proof instead of Lemma 3.4.) Analogous to Theorem 3.7, we have a result for the double proper nonnegative splitting. Theorem 4.7. Let A = P − R − S be a double proper nonnegative splitting of A ∈ Rm×n . Then the following are equivalent: (a) (b) (c) (d) (e) (f) (g)
ρ(W ) < 1; ρ(P Ď (R + S )) < 1; AĎ P − AĎ A ≥ O; AĎ P ≥ AĎ A; AĎ P ≥ P Ď (R + S ); AĎ (R + S ) ≥ O; [I − P Ď (R + S )]−1 ≥ O.
Comparison of spectral radii is very much useful for faster convergence of two different linear systems. In this direction Jena et al., [12] have obtained the following result. Theorem 4.8 (Theorem 3.1, [12]). Let A1 and A2 be two semi-monotone matrices having the same null space. Suppose that A1 = Ď Ď Ď Ď P1 − R1 + S1 and A2 = P2 − R2 + S2 be their double proper weak regular splittings. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. Being motivated by the above result, we now present a comparison result for double proper nonnegative splittings, and then obtain the same result for double nonnegative splittings of nonsingular matrices as a corollary. Theorem 4.9. Let A1 and A2 be two rectangular matrices having the same null space. Suppose that A1 = P1 − R1 − S1 and A2 = Ď Ď Ď Ď Ď Ď P2 − R2 − S2 be their double proper nonnegative splitting such that A1 P1 ≥ O and A2 P2 ≥ O. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. Proof. By Theorem 4.5, we have ρ(Wi ) < 1 for i = 1, 2. If ρ(W1 ) = 0, then our claim holds trivially. Suppose that ρ(W1 ) ̸= 0. Since A1 and A2 possess double proper nonnegative splitting, so W1 ≥ O and W2 ≥ O. Applying Theorem 2.3 to W1 , we get W1 x = ρ(W1 )x i.e., Ď
Ď
P1 R1 x1 + P1 S1 x2 = ρ(W1 )x1 x1 = ρ(W1 )x2 .
142
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144 Ď
Ď
Ď
Ď
Now N (A1 ) = N (A2 ) implies R(AT1 ) = R(AT2 ) = R(P1T ) = R(P2T ). Then P1 P1 = P2 P2 . The conditions P1 R1 ≥ P2 R2 and
(P2Ď R2
0 < ρ(W1 ) < 1 imply
W2 x − ρ(W1 )x =
Ď P1 R1 )x1
−
Ď
Ď
Ď
Ď
(P2Ď R2
Ď P1 R1 )x1 .
−
P2 R2 x1 + P2 S2 x2 − ρ(W1 )x1 x1 − ρ(W1 )x2
P2 R2 x1 + P2 S2 x2 − P1 R1 x1 − P1 S1 x2 0
=
=
≥
1 ρ(W1 )
Ď (P2 R2
−
Ď
Ď P1 R1 )x1
−
Ď
1
(P1Ď S1
ρ(W1 )
Therefore
−
Ď P2 S2 )x1
0
1
≥ ρ(W1 )
1
= ρ(W1 )
1
= ρ(W1 )
1
= ρ(W1 ) Ď
(P2Ď R2
Ď P1 R1 )x1
−
1
−
ρ(W1 )
0
(P1Ď S1
−
Ď P2 S2 )x1
[P2Ď (R2 + S2 ) − P1Ď (R1 + S1 )]x1 0
[P2Ď (P2
Ď P1 (P1
− A2 ) −
− A1 )]x1
0
[P1Ď A1 − P2Ď A2 ]x1
.
0
Ď
The condition P1 A1 ≥ P2 A2 then yields that W2 x − ρ(W1 )x ≥ 0 i.e., W2 x ≥ ρ(W1 )x. Thus, by Lemma 2.6, we have ρ(W1 ) ≤ ρ(W2 ) < 1. 1 Corollary 4.10. Suppose that A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 be their double nonnegative splitting such that A− 1 P1 ≥ O −1 −1 −1 −1 −1 and A2 P2 ≥ O. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1.
An example for the above result is shown next. Example 4.11. Let A1 =
3 0
0 3
0 0
, R1 =
5/6 1
2
−1
−1
3
0 0
0 0
0 0
and S1 =
and A2 = 1/6 0
1
−1
−1
2
0 0
. Then A1 and A2 have the same null space. Set P1 =
1 0
0 0
. Again, P2 =
2 0
0 2
0 0
, R2 =
1/5 1/5
0 0
0 0
Hence A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 are two double proper nonnegative splittings with Ď
Ď
Ď
Ď
We have P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 . Therefore 0.6379 = ρ(W1 ) ≤ ρ(W2 ) = 0.8924 < 1. Ď
Ď
Ď
and S2 =
Ď A1 P1
4/5 4/5
≥ 0 and
1 0
Ď A2 P2
0 0
.
≥ 0.
Ď
The example given below demonstrates that the conditions P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 cannot be dropped. Example 4.12. Let A1 =
1/3 3
0 0
0 0
and S1 =
2/3 0
2 −3 1 0
−1
2
0 0
0 0
. Again, P2 =
−1
1 −1
and A2 =
2 0
0 2
0 0
2
0 0
. Here N (A1 ) = N (A2 ). Set P1 =
, R2 =
1/2 1
0 0
0 0
Ď P2 − R2 − S2 are two double proper nonnegative splittings. Then P1 A1
and A2 = ρ(W1 ) ≥ ρ(W2 ) = 0.8689.
and S2 =
̸≥
Ď P2 A2
1/2 0 Ď
1 0
and P1 R1 ̸≥
3 0
0 2
0 0
, R1 =
0 0
. So A1 = P1 − R1 − S1
Ď P2 R2 , but 1
> 0.9293 =
The next example shows that the converse of Theorem 4.9 is not true.
Example 4.13. Let A1 =
−1 2
0 0
and A2 =
4 0
0 3
0 0
0 1
0 0
2/3 0
1 −1
−1 2
0 0
. Then A1 and A2 have the same null space. Set P1 =
2 0 0 1/2 0 0 1/2 1 0 , R1 = and S1 = . Again, P2 = 0 2 0 , R2 = 1 and S2 = 0 . 0 0 0 0 Therefore A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 are two double proper nonnegative splittings which satisfy the conditions Ď Ď Ď Ď Ď Ď A1 P1 ≥ 0 and A2 P2 ≥ 0. We then have 0.7829 = ρ(W1 ) ≤ ρ(W2 ) = 0.8689 < 1, but P1 A1 ̸≥ P2 A2 and P1 R1 ̸≥ P2 R2 .
4/3 1
2 −1
1 0
0 0
5. The group inverse analogue This section deals with the group inverse analogue of a few earlier results, and is meant for only square singular matrices. So, we assume all our matrices in this section are square matrices. Analogous to the definition of a proper nonnegative splitting, let us now introduce another definition for square singular matrices using the group inverse for studying the iterative scheme x(i+1) = U # Vx(i) + U # b.
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
143
Definition 5.1. A decomposition A = U − V of A ∈ Rn×n is called a proper G-nonnegative splitting if it is a proper splitting such that U # exists and U # V ≥ O. Recently, Mishra [19] has shown the following properties of a proper splitting using the group inverse of A. For completeness, we have reproduced the proof here. Theorem 5.2 (Theorem 3.4.1, [19]). Let A = U − V be a proper splitting of A ∈ Rn×n . Suppose that A# exists. Then (a) (b) (c) (d) (e)
U # exists. AA# = UU # ; A# A = U # U. A = U (I − U # V ). I − U # V is invertible. A# = (I − U # V )−1 U # .
Proof. (a) Since A = U − V is a proper splitting, so R(A) = R(U ) and N (A) = N (U ). Since A# exists, R(A) and N (A) are complementary subspaces and so are R(U ) and N (U ). Hence U # exists. (b) Since R(A) = R(U ) = R(A# ) and N (A) = N (U ) = N (U # ), it follows that PR(A),N (A) = PR(U ),N (U ) . Therefore AA# = UU # ; # A A = U # U. (c) Since R(A) = R(U ), it follows that R(V ) ⊆ R(A). Hence UU # V = V and then we have A = U − V = U − UU # V = U (I − U # V ). (d) Let (I − U # V )x = 0. Then x = U # Vx ∈ R(U # ) = R(U ) = R(A) and Ux = UU # Vx = Vx. So 0 = (U − V )x = Ax. We then have x = 0 since x ∈ R(A) ∩ N (A) which in turn implies the fact that I − U # V is invertible. (e) Set X = (I − U # V )−1 U # . Let x ∈ R(A). So x = A# Ax = U # Ux. Then XAx = (I − U # V )−1 U # (U − V )x = (I − U # V )−1 (U # Ux − U # Vx) = (I − U # V )−1 (x − U # Vx) = (I − U # V )−1 (I − U # V )x = x. Again, suppose that y ∈ N (A) = N (U ) = N (U # ). Then Xy = (I − U # V )−1 U # y = 0. This shows that A# = X = (I − U # V )−1 U # . A convergence result for the proper G-nonnegative splitting is shown next. The idea of this proof is very close to the idea of the proof of Lemma 3.4. ρ(A# U )−1
Lemma 5.3. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n and A# U ≥ O. Then ρ(U # V ) = ρ(A# U ) < 1. Proof. From the definition of a proper G-nonnegative splitting, we have A = U − V is a proper splitting with the conditions U # exists and U # V ≥ O. Then by Theorem 2.3, there exists a nonnegative vector x (x ̸= 0) such that U # Vx = ρ(U # V )x. Hence x ∈ R(U # ) = R(U ) = R(A) so that U # Ux = x. Since U # exists, so A# exists and A# = (I − U # V )−1 U # by Theorem 5.2(a) and (d). So A# U = (I − U # V )−1 U # U. Then
(I − U # V )−1 U # Ux = (I − U # V )−1 x =
1 1 − ρ(U # V )
x = A# Ux. ρ(A# U )−1
This implies 1−ρ(1U # V ) ≥ 0 and is an eigenvalue of A# U. Hence 0 ≤ 1−ρ(1U # V ) ≤ ρ(A# U ) i.e., ρ(U # V ) ≤ ρ(A# U ) . Again, the condition A# U ≥ O yields existence of a nonnegative vector y (y ̸= 0) such that A# Uy = ρ(A# U )y. Then y ∈ R(A) = R(U ). Therefore (I − U # V )−1 U # Uy = (I − U # V )−1 y = ρ(A# U )y. So, we have ρ(A#1 U )y = y − U # Vy i.e., U # Vy =
ρ(U V ) ≥ #
ρ(A# U )−1 . ρ(A# U )
Hence ρ(U V ) = #
ρ(A# U )−1 ρ(A# U )
< 1.
ρ(A# U )−1 y. ρ(A# U )
Thus
From the above proof it is clear that the proofs for all the remaining results can be done by using the similar idea as in case of the Moore–Penrose inverse and using the properties of group inverse of a matrix. Hence we present below a few other main results for square singular matrices without their proof. Lemma 5.4. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n . Then A# V ≥ O if and only if ρ(U # V ) = ρ(A# V ) 1+ρ(A# V )
< 1.
Theorem 5.5. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n . Then the following are equivalent:
ρ(U # V ) < 1; A# U − A# A ≥ O; A# U ≥ A# A; A# V ≥ U # V ; A# V ≥ O; ρ(A# V ) (f) ρ(U # V ) = 1+ρ(A# V ) .
(a) (b) (c) (d) (e)
With these results, let us move to the theory of double splittings for square singular matrices. At first, the definition of double proper G-nonnegative splitting is introduced below for index one matrices in connection with the iterative scheme xi+1 = P # Rxi + P # Sxi−1 + P # b.
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D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
Definition 5.6. A double decomposition A = P − R − S of A ∈ Rn×n is called double proper G-nonnegative splitting if A# exists, R(A) = R(P ), N (A) = N (P ), P # R ≥ O and P # S ≥ O. Note that the condition A# exists, R(A) = R(P ) and N (A) = N (P ) imply P # exists. A convergence theorem for the double proper G-nonnegative splittings is obtained below. Theorem 5.7. If A = P − R − S is a double proper G-nonnegative splitting with A# P ≥ O, then ρ(W ) < 1. Next one is the comparison result for two square singular linear systems. Theorem 5.8. Let A1 and A2 be two square singular matrices having the same null space. Suppose that A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 be their double proper G-nonnegative splittings, A#1 P1 ≥ O and A#2 P2 ≥ O. If P1# A1 ≥ P2# A2 and P1# R1 ≥ P2# R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. 6. Conclusions Many methods have been suggested in the literature to solve the square singular linear systems (1.1). The iterative methods (i.e., matrix splitting methods) and semi-iterative methods are among them. Recently, much attention is drawn to the Krylov subspace methods. It is known that the solution to a nonsingular linear system Ax = b lies in the Krylov space of (A, b)1 whose dimension is the degree of the minimal polynomial of A. However, the Krylov methods can fail if the matrix A is singular, and in some cases even if the linear system does have a solution, it may not lie in a Krylov space. Therefore in these cases, the theory of splittings may be useful. But the existence theorem of a Krylov solution for some square singular linear system says that a linear system Ax = b has a Krylov solution if and only if b ∈ R(Ak ), where k is the index of A. Another well-known result which relates the Krylov solution and the generalized inverse is described next. Let m be the degree of the minimal polynomial for A. If b ∈ R(Ak ), then the linear system Ax = b has a unique Krylov solution x = AD b ∈ Km−k (A, b). Otherwise, Ax = b does not have a solution in the Krylov space Kn (A, b), where n is the order of the matrix A. From the above results, it is clear that computation of AD b (or AD ) is important. This can be done by considering the iterative scheme xi+1 = U D Vxi + U D b, i = 0, 1, 2, . . . (or X i+1 = U D VX i + U D , i = 0, 1, 2, . . . , where X denotes a matrix). The convergence of the above iterative scheme will go in a very similar way as followed in this article and will be studied in future. Acknowledgements The author thanks the anonymous referees for their valuable suggestions which improved the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
R.S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. Y. Song, Comparisons of nonnegative splittings of matrices, Linear Algebra Appl. 154–156 (1991) 453–455. I. Marek, D.B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math. 56 (1989) 283–289. Y. Song, Comparison theorems for splittings of matrices, Numer. Math. 92 (2002) 563–591. J. Song, Y. Song, Convergence for nonnegative double splittings of matrices, Calcolo 48 (2011) 245–260. Z.I. Woźnicki, Nonnegative splitting theory, Japan J. Indust. Appl. Math. 11 (1994) 289–342. G. Csordas, R.S. Varga, Comparisons of regular splittings of matrices, Numer. Math. 44 (1984) 23–35. L. Elsner, Comparisons of weak regular splittings and multi splitting methods, Numer. Math. 56 (1989) 283–289. Z.I. Woźnicki, Matrix splitting principles, Novi Sad J. Math. 28 (1998) 197–209. A. Berman, R.J. Plemmons, Cones and iterative methods for best square least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974) 145–154. L. Jena, D. Mishra, S. Pani, Convergence and comparisons of single and double decompositions of rectangular matrices, Calcolo (2013). http://dx.doi. org/10.1007/s10092-013-0079-3. Z.I. Woźnicki, Estimation of the optimum relaxation factors in partial factorization iterative methods, SIAM J. Matrix Anal. Appl. 14 (1993) 59–73. S.-Q. Shen, T.-Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl. 51 (2006) 1751–1760. S.-X. Miao, B. Zheng, A note on double splittings of different monotone matrices, Calcolo 46 (2009) 261–266. A. Ben-Israel, T.N.E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003. D. Mishra, K.C. Sivakumar, On splittings of matrices and nonnegative generalized inverses, Oper. Matrices 6 (1) (2012) 85–95. D. Mishra, K.C. Sivakumar, Comparison theorems for a subclass of proper splittings of matrices, Appl. Math. Lett. 25 (2012) 2339–2343. D. Mishra, Least elements, matrix splittings and nonnegative generalized inverses, Ph.D. Thesis, IIT Madras, 2012.
1 K (A, b) = span{b, Ab, A2 b, . . . , As−1 b}, where s denotes the dimension. s
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Nonnegative splittings for rectangular matrices Debasisha Mishra Department of Mathematics, National Institute of Technology Raipur, Raipur- 492010, India
article
info
Article history: Received 11 March 2013 Received in revised form 17 September 2013 Accepted 26 October 2013 Keywords: Moore–Penrose inverse Linear system Nonnegative splitting Proper splitting Convergence theorem Comparison theorem
abstract The extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first. Different convergence and comparison theorems for the proper nonnegative splittings are established. The notion of double nonnegative splitting is then generalized to rectangular matrices. Finally, different convergence and comparison results are presented for this decomposition. The case for singular square matrices is also studied. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The theory of splittings for square nonsingular matrices and its relationship with the solution of system of linear equations is quite well-known. Standard iterative methods like the Jacobi, Gauss–Seidel and successive over-relaxation methods for solving a square nonsingular system of linear equations Ax = b, arise from different choices of real square matrices U and V , where A = U − V and b is a real n-vector. The book by Varga [1] contains several splittings such as regular and weak regular splittings. A decomposition A = U − V of a real square nonsingular matrix A is (i) regular splitting if U −1 exists, U −1 ≥ O and V ≥ O [1], (ii) weak regular splitting if U −1 exists, U −1 ≥ O and U −1 V ≥ O [2,1], (iii) nonnegative if U −1 exists and U −1 V ≥ O [3], where the comparison is entrywise and O is the null matrix. The theory of nonnegative splittings is analyzed in [4,3,5–7]. Also, Csordas and Varga [8], Elsner [9], Song [3,5], Song and Song [6], Woźnicki [10] and many others have proved various comparison results for different matrix splittings. Berman and Plemmons [11] then extended the concept of splittings to rectangular matrices and called it as a proper splitting. A decomposition A = U − V of A ∈ Rm×n is called a proper splitting [11] if R(A) = R(U ) and N (A) = N (U ), where R(A) and N (A) denote the range space and the null space of A. Linear systems of the form Ax = b,
(1.1)
where A is a real square singular or real sparse or real rectangular matrix appear in many areas of mathematics. For example rectangular/singular systems arise by applying finite difference methods to partial differential equations such as the Neumann Problem and Poisson’s equation. The iteration x(i+1) = U Ď Vx(i) + U Ď b, E-mail address: [email protected]. 0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.10.012
(1.2)
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
137
is often employed to solve these systems, where BĎ means the Moore–Penrose inverse of B (see next section for the definition). The scheme (1.2) is said to be convergent if the spectral radius of U Ď V is less than 1, and U Ď V is called the iteration matrix. The authors of [11] showed that if A = U − V is a proper splitting, then the scheme (1.2) converges to AĎ b, the least square solution of minimum norm for any initial vector x0 if and only if the spectral radius of U Ď V is less than 1. (See Corollary 1, [11].) However, it is not true for any initial vector x0 . As if x0 = 0 or x0 ∈ N (V ), then the iterative sequence will not move further. Hence the initial vector x0 should not be a zero-vector and should not lie in the null space of V . (Interested readers can have a look at the introductory part of the article [11] for the reason and importance of choosing proper splittings.) We also remark that the scheme Y (j+1) = U Ď VY (j) + U Ď also converges to AĎ under analogous conditions for a suitable initial matrix Y 0 . Very recently, the authors of [12] extended the notion of regular and weak regular splittings to rectangular matrices and the respective definitions are recalled next. A decomposition A = U − V of A ∈ Rm×n is called a proper regular splitting if it is a proper splitting such that U Ď ≥ O and V ≥ O. Similarly, A = U − V is called proper weak regular splitting if it is a proper splitting such that U Ď ≥ 0 and U Ď V ≥ O. Note that Berman and Plemmons [11] proved a convergence theorem for these splittings without specifying the types of matrix decompositions. Using the notion of proper regular splitting, Theorem 3, [11] can be now rewritten as follows. Let A = U − V be a proper regular splitting of A ∈ Rm×n . Then AĎ ≥ O if and only if ρ(U Ď V ) < 1. In [12], one can find comparison results for these splittings and their applications to the double splitting theory. But, in this paper, our plan is to introduce another new decomposition which is an extension of so-called nonnegative splitting for square nonsingular matrices, and is more general than the proper regular and proper weak regular splittings (see Section 3 for more discussion). Eq. (1.1) can also be solved using double decomposition of A. (A decomposition of a real m × n matrix of the form A = P −R−S is called double decomposition; for the real square nonsingular case it is called double splitting if P is nonsingular.) The idea of double splitting was first introduced by Woźnicki [13] for square nonsingular matrices. This notion was then extended by Jena et al. [12] for real m × n matrices. Shen et al. [14], Miao and Zheng [15] and Song and Song [6] have studied convergence and comparison theorems of real square nonsingular matrices using double splittings. In particular, Song and Song [6] proved that the double splitting is convergent if and only if the single splitting is convergent for the nonnegative splittings. In this paper, we are going to extend the same result for real rectangular matrices along with a comparison result. The central idea of this paper is to study the theory of nonnegative splittings for rectangular matrices. The organization is as follows. In Section 2, we list all relevant definitions, notation and some earlier results which we use in the paper. The main results are given in Sections 3 and 4. Section 3 introduces the generalization of nonnegative splitting to rectangular matrices, and then discusses convergence and comparison theorems for these decompositions. In Section 4, we propose the notion of double proper nonnegative splittings for real m × n matrices. Then convergence results for double proper nonnegative splittings are established. At last, we obtain a comparison theorem for two different linear systems. Section 5 discusses the group inverse analogue of a few main results mentioned in Sections 3 and 4 for square singular matrices. Finally, we end up with conclusions. 2. Preliminaries Let Rn denote the n dimensional real Euclidean space and Rn+ denote the nonnegative orthant in Rn . For a real m × n matrix A, i.e., A ∈ Rm×n , the matrix G satisfying the four equations known as Penrose equations: AGA = A, GAG = G, (AG)T = AG and (GA)T = GA is called the Moore–Penrose inverse of A (BT denotes the transpose of B). It always exists and is unique, and is denoted by AĎ . A ∈ Rm×n is said to be semi-monotone if AĎ ≥ O. The group inverse of a matrix A ∈ Rn×n (if it exists), denoted by A# is the unique matrix X satisfying A = AXA, X = XAX and AX = XA. Equivalently, A# is the unique matrix X which satisfies XAx = x for all x ∈ R(A) and Xy = 0 for all y ∈ N (A). The index of a real square matrix A is the least nonnegative integer k such that rank(Ak+1 ) = rank(Ak ). It is well known that A# exists if and only if index of A is 1 (i.e., R(A) ⊕ N (A) = Rn ). Let A ∈ Rn×n be of index k. Then, the Drazin inverse of A is the unique matrix AD ∈ Rn×n which satisfies the equations Ak+1 AD = Ak , AD AAD = AD and AAD = AD A. A ∈ Rn×n is said to be group monotone if A# exists and A# ≥ O. Similarly, it is called Drazin monotone if AD ≥ O. When A is a square nonsingular, then AĎ = A−1 = A# = AD , and a semi-monotone matrix (as well as a group monotone matrix or a Drazin monotone matrix) becomes a monotone matrix (i.e., A−1 exists and A−1 ≥ O). (See the book by Berman and Plemmons, [2] for more details on monotone matrices and their generalizations.) For A, B, C ∈ Rm×n , we say A is nonnegative if A ≥ O, and B ≥ C if B − C ≥ O. We denote a nonnegative vector x as x ≥ 0. Let K , L be complementary subspaces of Rp , i.e., K ⊕ L = Rp . Then PK ,L denotes the (not necessarily orthogonal) projection of Rp onto K along L. Thus PK2,L = PK ,L , R(PK ,L ) = K and N (PK ,L ) = L. If in addition, K ⊥ L, PK ,L will be denoted by PK . A few properties of AĎ and A# [16] are listed here: R(AT ) = R(AĎ ); N (AT ) = N (AĎ ); AAĎ = PR(A) ; AĎ A = PR(AT ) ; R(A) = R(A# ); N (A) = N (A# ); AA# = PR(A),N (A) . In particular, if x ∈ R(A∗ ) then x = AĎ Ax and if x ∈ R(A) then x = A# Ax. The spectral radius of A ∈ Rn×n , denoted by ρ(A) is defined by ρ(A) = max1≤i≤n |λi |, where λ1 , λ2 , . . . , λn are the eigenvalues of A. The following results will be helpful to prove our main result. The first one is a part of Theorem 1, [11] and Theorem 3.1, [17] which expresses A and AĎ in terms of U and V using the proper splitting A = U − V . Theorem 2.1. Let A = U − V be a proper splitting. Then (a) AAĎ = UU Ď ; AĎ A = U Ď U.
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D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
(b) A = U (I − U Ď V ) = (I − VU Ď )U. (c) I − U Ď V and I − VU Ď are invertible. (d) AĎ = (I − U Ď V )−1 U Ď = U Ď (I − VU Ď )−1 . The next lemma shows a relation between the eigenvalues of U Ď V and AĎ V . Lemma 2.2 (Lemma 2.6, [18]). Let A = U − V be a proper splitting of A ∈ Rm×n . Let µi , 1 ≤ i ≤ s and λj , 1 ≤ j ≤ s be the eigenvalues of the matrices U Ď V (VU Ď ) and AĎ V (VAĎ ) respectively. Then for every j, we have 1 + λj ̸= 0. Also, for every i, there λ
µ
j i and for every j, there exists i such that λj = 1−µ . exists j such that µi = 1+λ j i
The next theorem is a part of the Perron–Frobenius theorem. Theorem 2.3 (Theorem 2.20, [1]). Let A ≥ O. Then A has a nonnegative real eigenvalue equal to its spectral radius. Another result which relates spectral radius of two nonnegative matrices is given below. Theorem 2.4 (Theorem 2.21, [1]). Let A ≥ B ≥ O. Then ρ(A) ≥ ρ(B). In the theory of nonnegative matrices, the following three results are well known. Theorem 2.5 (Theorem 3.16, [1]). Let X ∈ Rn×n and X ≥ O. Then ρ(X ) < 1 if and only if (I − X )−1 exists and (I − X )−1 = ∞ k k=0 X ≥ O. Lemma 2.6 (Corollary 3.2, [4]). Let B ≥ O and x ≥ 0 be such that Bx − α x ≥ 0. Then α ≤ ρ(B). Lemma 2.7 (Lemma 2.2, [14]). Let X =
B I
C O
≥ O and ρ(B + C ) < 1. Then ρ(X ) < 1.
3. Proper nonnegative splittings Let us begin this section with the definition of a Z -matrix and an M-matrix. A square matrix whose off-diagonal elements are non-positive is called a Z -matrix. A Z -matrix A is called a nonsingular M-matrix if A is monotone. Equivalently, a Z -matrix A is called a nonsingular M-matrix if A = sI − B, where s > ρ(B) and B ≥ O. More on M-matrices can be found in Chapter 6 of [2]. Setting U = sI and V = B, we always have a nonnegative splitting. The problem is when s = ρ(B) and in that case A becomes a singular M-matrix. The Laplacian matrix of an undirected graph is a singular M-matrix. To deal with this situation and also in more general way i.e., the case when A is a real rectangular matrix, we propose the following definition. Definition 3.1. A decomposition A = U − V of A ∈ Rm×n is called a proper nonnegative splitting if it is a proper splitting such that U Ď V ≥ O. The class of matrices having a positive real number in all entries always have proper nonnegative splittings. We also present another two classes of matrices which always have the same decomposition. Example 3.2. Let A ∈ Rn×n and be of the form (n−r )×r
(n−r )×(n−r )
B O2
O1 O3
where B is a nonsingular M-matrix of order r , O1 ∈ Rr ×(n−r ) , O2 ∈
R and O3 ∈ R are zero matrices. Then A is a singular Z -matrix. Take U = diag(b11 , b22 , . . . , brr , 0, 0, . . . , 0) where B = (bij ) and V = U − A. Then A = U − V is a proper nonnegative splitting. Example 3.3. Let A ∈ R2n×n be partitioned as A = U =
D1 D2
B C
where B and C are real n × n nonsingular M-matrices. Setting
, where D1 = diag(b11 , b22 , . . . , bnn ) and D2 = diag(c11 , c22 , . . . , cnn ) with B = (bij ) and C = (cij ), respectively.
Then V = U − A and thus we have A = U − V is a proper nonnegative splitting. We now present a convergence result for the proper nonnegative splitting which is an extension of Lemma 2.1, [3] for real rectangular matrices. Note that the converse of the result given below is not true in general. Nevertheless, this is true by assumption of one more condition AĎ A ≥ O. ρ(AĎ U )−1
Lemma 3.4. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n and AĎ U ≥ O. Then ρ(U Ď V ) = ρ(AĎ U ) < 1. Proof. Since A = U − V is a proper nonnegative splitting, so A = U − V is a proper splitting with U Ď V ≥ O. By Theorem 2.3, there exists a nonnegative vector x (x ̸= 0) such that U Ď Vx = ρ(U Ď V )x. Hence x ∈ R(U Ď ) = R(U T ) = R(AT ) so that U Ď Ux = x. By Theorem 2.1(d), we also have AĎ = (I − U Ď V )−1 U Ď . So AĎ U = (I − U Ď V )−1 U Ď U. Then
(I − U Ď V )−1 U Ď Ux = (I − U Ď V )−1 x =
1 1 − ρ(U Ď V )
x = AĎ Ux. ρ(AĎ U )−1
This implies 1−ρ(1U Ď V ) ≥ 0 and is an eigenvalue of AĎ U. Hence 0 ≤ 1−ρ(1U Ď V ) ≤ ρ(AĎ U ) i.e., ρ(U Ď V ) ≤ ρ(AĎ U ) . Again, the condition AĎ U ≥ O implies existence of a nonnegative vector y (y ̸= 0) such that AĎ Uy = ρ(AĎ U )y. Then y ∈ R(AT ) = R(U T ).
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
Therefore (I − U Ď V )−1 U Ď Uy = (I − U Ď V )−1 y = ρ(AĎ U )y. So, we have ρ(AĎ1U )y = y − U Ď Vy i.e., U Ď Vy = Ď
ρ(U V ) ≥
ρ(AĎ U )−1 . ρ(AĎ U )
Ď
Hence ρ(U V ) =
ρ(AĎ U )−1 ρ(AĎ U )
< 1.
139
ρ(AĎ U )−1 ρ(AĎ U )
y. Thus
Ď j Ď Since ∀ x ∈ Rn+ ∩ R(AT ), AĎ Ux = (I − U Ď V )−1 U Ď Ux = (I − U Ď V )−1 x = j=0 (U V ) x ≥ 0 (if ρ(U V ) < 1). Hence the n T converse of the above theorem is also true on R+ ∩ R(A ). Another convergence result for the proper nonnegative splitting is obtained below. A similar proof as in the above theorem can be provided. However, we present a new proof to this.
∞
ρ(AĎ V )
Lemma 3.5. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n . Then AĎ V ≥ O if and only if ρ(U Ď V ) = 1+ρ(AĎ V ) < 1. Proof. We have AĎ V ≥ O and U Ď V ≥ O. Let λ and µ be any nonnegative eigenvalues of AĎ V and U Ď V , respectively. Let λ λ f (λ) = 1+λ , λ ≥ 0. Then f is a strictly increasing function. Then by Lemma 2.2, µ = 1+λ . So, µ attains its maximum when
λ is maximum. But λ is maximum when λ = ρ(AĎ V ). As a result, the maximum value of µ is ρ(U Ď V ). Hence, ρ(U Ď V ) = ρ(AĎ V ) < 1. 1+ρ(AĎ V ) Conversely, A = U − V is splitting yields AĎ =(I − U Ď V )−1 U Ď . By Theorem 2.5, the condition ρ(U Ď V ) < 1 a∞proper ∞ Ď k Ď k ( U V ) . Therefore AĎ V = implies that (I − U Ď V )−1 = k=1 (U V ) ≥ O. k=0 The above lemma admits the following corollary in case of nonsingular matrices. Corollary 3.6 (Lemma 2.2, [3]). Let A = U − V be a nonnegative splitting of A ∈ Rn×n . Then A−1 V ≥ O if and only if ρ(U −1 V ) = ρ(A−1 V ) 1+ρ(A−1 V )
< 1.
We then present certain properties of proper nonnegative splitting. Theorem 3.7. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n and AĎ V ≥ O. Then (a) (I − U Ď V )−1 ≥ O. (b) (I − U Ď V )−1 ≥ I. (c) AĎ V ≥ U Ď V . Proof. (a) By Lemma 3.5, we have ρ(U Ď V ) < 1. Then, Theorem 2.5 implies that (I − U Ď V )−1 exists and (I − U Ď V )−1 = ∞ Ď k condition U Ď V ≥ O implies (I − U Ď V )−1 ≥ O. k=0 (U V ) . Finally the ∞ ∞ Ď k Ď k (b) (I − U Ď V )−1 = k=0 (U V ) = I + k=1 (U V ) . As the decomposition is proper nonnegative, so the second term in the above expression is nonnegative. Hence the claim follows. (c) From Theorem 2.1(d), we have AĎ = (I − U Ď V )−1 U Ď . So (I − U Ď V )AĎ = U Ď . Then post-multiplying by V , we have AĎ V − U Ď V = U Ď VAĎ V . Since AĎ V and U Ď V are nonnegative, so AĎ V − U Ď V ≥ O. Hence AĎ V ≥ U Ď V . The next theorem generalizes some parts of Lemma 2.6, [5] to rectangular matrices. Theorem 3.8. Let A = U − V be a proper nonnegative splitting of A ∈ Rm×n . Then the following are equivalent:
ρ(U Ď V ) < 1; AĎ U − AĎ A ≥ O; AĎ U ≥ AĎ A; AĎ V ≥ U Ď V ; AĎ V ≥ O; ρ(AĎ V ) (f) ρ(U Ď V ) = 1+ρ(AĎ V ) .
(a) (b) (c) (d) (e)
Proof. ⇒ (b) A = U − V is a proper splitting implies AĎ = (I − U Ď V )−1 U Ď . Again ρ(U Ď V ) < 1 yields that (I − U Ď V )−1 = ∞ (a) Ď T T Ď T T Ď ( U V )k . Since k=0 ∞ N (A) = N (U ) so N (A) ⊆ N (V ) i.e. R(V ) ⊆ R(A ). Hence U UV = V which gives VU U = V . Therefore AĎ U = U Ď U + k=1 (U Ď V )k ≥ O. The fact U Ď V ≥ O implies that AĎ U − AĎ A ≥ O. (b) ⇒ (c) Clearly. (c) ⇒ (d) Since U Ď V ≥ O, post-multiplying U Ď V to AĎ U ≥ AĎ A we get AĎ UU Ď V ≥ AĎ AU Ď V = U Ď UU Ď V . Also, the condition R(A) = R(U ) implies R(V ) ⊆ R(A) = R(U ). Hence UU Ď V = V . We thus have AĎ V ≥ U Ď V . (d) ⇒ (e) Since U Ď V ≥ O, so AĎ V ≥ U Ď V ≥ O. (e) ⇒ (f) From Lemma 3.5. (f) ⇒ (a) Obviously. Another result for a convergent proper nonnegative splitting is shown next. Theorem 3.9. Let A = U − V be a convergent proper nonnegative splitting of A ∈ Rm×n . If AĎ ≥ O, then (a) AĎ ≥ U Ď ; (b) AĎ V ≥ U Ď V ;
ρ(AĎ V )
(c) ρ(U Ď V ) = 1+ρ(AĎ V ) .
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D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
The proof of (c) is analogous to the proof of sufficient part of Lemma 3.5, while (a) and (b) follows using the properties of a proper splittings. The condition AĎ V2 ≥ AĎ V1 ≥ O gives ρ(AĎ V2 ) ≥ ρ(AĎ V1 ) by Theorem 2.4. Again, using the same idea as in the proof of Lemma 3.5, we have the following comparison theorem for two proper nonnegative splittings. Theorem 3.10. Let A = U1 − V1 = U2 − V2 be two proper nonnegative splittings of A ∈ Rm×n . If AĎ V2 ≥ AĎ V1 ≥ O, then Ď
Ď
1 > ρ(U2 V2 ) ≥ ρ(U1 V1 ). The next result is the final result of this section, and is a generalization of Theorem 2.17, [5]. The proof is analogous to nonsingular case and thus omitted. Theorem 3.11. Let A = U1 − V1 = U2 − V2 be two convergent proper nonnegative splittings of A ∈ Rm×n . Let x ≥ 0 and z ≥ 0 be Ď Ď Ď Ď Ď such that U1 V1 x = ρ(U1 V1 )x and U2 V2 z = ρ(U2 V2 )z. Suppose that AĎ ≥ O and either V1 x ≥ 0 with ρ(U1 V1 )x > 0 or V2 z ≥ 0 Ď
Ď
Ď
with z > 0 and ρ(U2 V2 )z > 0. Further, assume that there exists α, 0 < α ≤ 1 such that U2 ≤ α U1 . Then
ρ(U1Ď V1 ) ≤ ρ(U2Ď V2 ) < 1,
when α = 1 and
ρ(U1Ď V1 )
when 0 < α < 1.
ρ(U2Ď V2 )
<
< 1,
A decomposition A = U − V of A ∈ Rm×n is called proper nonnegative splitting of 2nd type if it is a proper splitting with VU Ď ≥ O. In case of nonsingular matrices, this is called as nonnegative splitting of 2nd type. We also mention that proper nonnegative splitting of 1st type is same as proper nonnegative splitting. In the literature, these types of splittings have different names. However, we will use these names. Now, we conclude this section with the remark that similar results for the proper nonnegative splitting of 2nd type can also be obtained with some minor changes. 4. Double proper nonnegative splittings Analogous to the idea of Jena et al. [12] the double decomposition A = P − R − S of A ∈ Rm×n to (1.1) leads to the following iterative scheme spanned in three iterates: xi+1 = P Ď Rxi + P Ď Sxi−1 + P Ď b,
i > 0.
(4.1)
Then
xi+1 xi
where yi+1 =
=
P ĎR I
i+1
x
i
x
P ĎS O
, yi =
xi
i −1
x xi
i−1
x
+
,W =
P Ďb , O
PĎR I
PĎS O
and d =
PĎb O
i.e.,
yi+1 = Wyi + d.
(4.2)
Then the above scheme is convergent if ρ(W ) < 1. In this case, we say A = P − R − S is a convergent double decomposition. Jena et al. [12] introduced two double decompositions called double proper regular splitting and double proper weak regular splitting. A double decomposition A = P − R + S is called double proper regular splitting if R(A) = R(P ), N (A) = N (P ), P Ď ≥ O, R ≥ O and S ≤ O. A double decomposition A = P − R + S is called double proper weak regular splitting if R(A) = R(P ), N (A) = N (P ), P Ď ≥ 0, P Ď R ≥ O and P Ď S ≤ O. Note that Jena et al. considered the decomposition A = P − R + S while we are considering A = P − R − S. The definition of a new double decomposition called double proper nonnegative splitting is now proposed. Definition 4.1. A double decomposition A = P − R − S of A ∈ Rm×n is called double proper nonnegative splitting if R(A) = R(P ), N (A) = N (P ), P Ď R ≥ O and P Ď S ≥ O. In case of nonsingular matrices, we have the following definition. Definition 4.2 (Definition 1.3, [6]). A double decomposition A = P − R − S of A ∈ Rn×n is called double nonnegative splitting if P −1 R ≥ O and P −1 S ≥ O. Setting P = U and R − S = V in Definition 4.1, we obtain proper nonnegative splittings. We first prove a convergence result which relates convergence of single and double decompositions. Theorem 4.3. Let A = P − R − S be a double proper nonnegative splitting of A ∈ Rm×n . Then ρ(W ) < 1 if and only if ρ(U Ď V ) < 1, where U = P and V = R + S. Proof. Since A = P − R − S be a double proper nonnegative splitting, so W ≥ O. Then, Theorem 2.5 says that ρ(W ) < 1 if and only if (I − W )−1 exists and (I − W )−1 ≥ O. A = P − (R + S ) = U − V is a proper splitting implies that [I − P Ď (R + S )]−1
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
141
exists (by Theorem 2.1(c)). By block matrix computation, we get
(I − W )
−1
=
[I − P Ď (R + S )]−1 O
=
Ď
[I − P (R + S )]−1 [I − P Ď (R + S )]−1
O [I − P Ď (R + S )]−1
I I
P ĎS I − P ĎR
[I − P Ď (R + S )]−1 P Ď S . [I − P Ď (R + S )]−1 (I − P Ď R)
If (I − W )−1 ≥ O, then [I − P Ď (R + S )]−1 ≥ O. By Theorem 2.5, we get ρ(U Ď V ) = ρ(P Ď (R + S )) < 1. Ď Ď −1 −1 Conversely, if ρ(U Ď V ) = ρ(P (R1 + S )) < 1, then [I − P (R + S )] ≥ O. We will now show that (I − W ) exists. Let x I − PĎR −P Ď S 1 2 T Ď 1 Ď 2 1 2 (I − W )x = 0. So −I = 0, where x = (x , x ) . Then, we have I − P Rx = P Sx and x = x . For simplicity 2 I x
put x1 = x2 = y. So y = (P Ď R + P Ď S )y = P Ď (R + S )y = U Ď Vy. Hence y ∈ R(U T ) = R(AT ). Again, the condition y = U Ď Vy implies Uy = UU Ď Vy = Vy since we have R(V ) ⊆ R(A) (which follows from the condition R(A) = R(U )). Thus Uy = Vy which yields (U − V )y = Ay = 0. Therefore we get x = 0. Hence I − W is invertible. ∞ y ∈ĎN (A) which in∞turnĎ impliesj yĎ= 0 and then j Ď Also [I − P Ď (R + S )]−1 (I − P Ď R) = ( P ( R + S )) − ( P ( R + S )) P R = I + P ( R + S )− P Ď R +[P Ď (R + S )]2 − P Ď (R + S )P Ď R + j=0 j=0
· · · = I + P Ď S + P Ď (R + S )(P Ď S ) + [P Ď (R + S )]2 (P Ď S ) + · · · ≥ I ≥ O since P Ď R and P Ď S are nonnegative. The condition [I − P Ď (R + S )]−1 (I − P Ď R) ≥ O implies (I − W )−1 ≥ O. Hence ρ(W ) < 1. As a corollary to the above result Theorem 2.2, [6] is obtained and is shown next. Corollary 4.4 (Theorem 2.2, [6]). Let A = P − R − S be a double nonnegative splitting of A ∈ Rn×n . Then ρ(W ) < 1 if and only if ρ(U −1 V ) < 1, where U = P and V = R + S. We then present a convergence theorem for the double proper nonnegative splittings. Thereafter a similar result is obtained for the double nonnegative splitting as a corollary. Theorem 4.5. Let AĎ P ≥ O. If A = P − R − S is a double proper nonnegative splitting of A ∈ Rm×n , then ρ(W ) < 1. Proof. A = P − R − S is a double proper nonnegative splitting implies W =
PĎR I
PĎS O Ď
≥ O. Setting U = P and V = R + S,
we get A = U − V is a proper nonnegative splitting. Then by Lemma 3.4, we have ρ(P (R + S )) = ρ(U Ď V ) < 1. By Lemma 2.7, it now follows that ρ(W ) < 1. Corollary 4.6. Let A−1 P ≥ O. If A = P − R − S is a double nonnegative splitting of A ∈ Rn×n , then ρ(W ) < 1. Theorem 4.5 is also true if the condition AĎ P ≥ O is replaced by AĎ (R + S ) ≥ O. (Then Lemma 3.5 will be used in the proof instead of Lemma 3.4.) Analogous to Theorem 3.7, we have a result for the double proper nonnegative splitting. Theorem 4.7. Let A = P − R − S be a double proper nonnegative splitting of A ∈ Rm×n . Then the following are equivalent: (a) (b) (c) (d) (e) (f) (g)
ρ(W ) < 1; ρ(P Ď (R + S )) < 1; AĎ P − AĎ A ≥ O; AĎ P ≥ AĎ A; AĎ P ≥ P Ď (R + S ); AĎ (R + S ) ≥ O; [I − P Ď (R + S )]−1 ≥ O.
Comparison of spectral radii is very much useful for faster convergence of two different linear systems. In this direction Jena et al., [12] have obtained the following result. Theorem 4.8 (Theorem 3.1, [12]). Let A1 and A2 be two semi-monotone matrices having the same null space. Suppose that A1 = Ď Ď Ď Ď P1 − R1 + S1 and A2 = P2 − R2 + S2 be their double proper weak regular splittings. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. Being motivated by the above result, we now present a comparison result for double proper nonnegative splittings, and then obtain the same result for double nonnegative splittings of nonsingular matrices as a corollary. Theorem 4.9. Let A1 and A2 be two rectangular matrices having the same null space. Suppose that A1 = P1 − R1 − S1 and A2 = Ď Ď Ď Ď Ď Ď P2 − R2 − S2 be their double proper nonnegative splitting such that A1 P1 ≥ O and A2 P2 ≥ O. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. Proof. By Theorem 4.5, we have ρ(Wi ) < 1 for i = 1, 2. If ρ(W1 ) = 0, then our claim holds trivially. Suppose that ρ(W1 ) ̸= 0. Since A1 and A2 possess double proper nonnegative splitting, so W1 ≥ O and W2 ≥ O. Applying Theorem 2.3 to W1 , we get W1 x = ρ(W1 )x i.e., Ď
Ď
P1 R1 x1 + P1 S1 x2 = ρ(W1 )x1 x1 = ρ(W1 )x2 .
142
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144 Ď
Ď
Ď
Ď
Now N (A1 ) = N (A2 ) implies R(AT1 ) = R(AT2 ) = R(P1T ) = R(P2T ). Then P1 P1 = P2 P2 . The conditions P1 R1 ≥ P2 R2 and
(P2Ď R2
0 < ρ(W1 ) < 1 imply
W2 x − ρ(W1 )x =
Ď P1 R1 )x1
−
Ď
Ď
Ď
Ď
(P2Ď R2
Ď P1 R1 )x1 .
−
P2 R2 x1 + P2 S2 x2 − ρ(W1 )x1 x1 − ρ(W1 )x2
P2 R2 x1 + P2 S2 x2 − P1 R1 x1 − P1 S1 x2 0
=
=
≥
1 ρ(W1 )
Ď (P2 R2
−
Ď
Ď P1 R1 )x1
−
Ď
1
(P1Ď S1
ρ(W1 )
Therefore
−
Ď P2 S2 )x1
0
1
≥ ρ(W1 )
1
= ρ(W1 )
1
= ρ(W1 )
1
= ρ(W1 ) Ď
(P2Ď R2
Ď P1 R1 )x1
−
1
−
ρ(W1 )
0
(P1Ď S1
−
Ď P2 S2 )x1
[P2Ď (R2 + S2 ) − P1Ď (R1 + S1 )]x1 0
[P2Ď (P2
Ď P1 (P1
− A2 ) −
− A1 )]x1
0
[P1Ď A1 − P2Ď A2 ]x1
.
0
Ď
The condition P1 A1 ≥ P2 A2 then yields that W2 x − ρ(W1 )x ≥ 0 i.e., W2 x ≥ ρ(W1 )x. Thus, by Lemma 2.6, we have ρ(W1 ) ≤ ρ(W2 ) < 1. 1 Corollary 4.10. Suppose that A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 be their double nonnegative splitting such that A− 1 P1 ≥ O −1 −1 −1 −1 −1 and A2 P2 ≥ O. If P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1.
An example for the above result is shown next. Example 4.11. Let A1 =
3 0
0 3
0 0
, R1 =
5/6 1
2
−1
−1
3
0 0
0 0
0 0
and S1 =
and A2 = 1/6 0
1
−1
−1
2
0 0
. Then A1 and A2 have the same null space. Set P1 =
1 0
0 0
. Again, P2 =
2 0
0 2
0 0
, R2 =
1/5 1/5
0 0
0 0
Hence A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 are two double proper nonnegative splittings with Ď
Ď
Ď
Ď
We have P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 . Therefore 0.6379 = ρ(W1 ) ≤ ρ(W2 ) = 0.8924 < 1. Ď
Ď
Ď
and S2 =
Ď A1 P1
4/5 4/5
≥ 0 and
1 0
Ď A2 P2
0 0
.
≥ 0.
Ď
The example given below demonstrates that the conditions P1 A1 ≥ P2 A2 and P1 R1 ≥ P2 R2 cannot be dropped. Example 4.12. Let A1 =
1/3 3
0 0
0 0
and S1 =
2/3 0
2 −3 1 0
−1
2
0 0
0 0
. Again, P2 =
−1
1 −1
and A2 =
2 0
0 2
0 0
2
0 0
. Here N (A1 ) = N (A2 ). Set P1 =
, R2 =
1/2 1
0 0
0 0
Ď P2 − R2 − S2 are two double proper nonnegative splittings. Then P1 A1
and A2 = ρ(W1 ) ≥ ρ(W2 ) = 0.8689.
and S2 =
̸≥
Ď P2 A2
1/2 0 Ď
1 0
and P1 R1 ̸≥
3 0
0 2
0 0
, R1 =
0 0
. So A1 = P1 − R1 − S1
Ď P2 R2 , but 1
> 0.9293 =
The next example shows that the converse of Theorem 4.9 is not true.
Example 4.13. Let A1 =
−1 2
0 0
and A2 =
4 0
0 3
0 0
0 1
0 0
2/3 0
1 −1
−1 2
0 0
. Then A1 and A2 have the same null space. Set P1 =
2 0 0 1/2 0 0 1/2 1 0 , R1 = and S1 = . Again, P2 = 0 2 0 , R2 = 1 and S2 = 0 . 0 0 0 0 Therefore A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 are two double proper nonnegative splittings which satisfy the conditions Ď Ď Ď Ď Ď Ď A1 P1 ≥ 0 and A2 P2 ≥ 0. We then have 0.7829 = ρ(W1 ) ≤ ρ(W2 ) = 0.8689 < 1, but P1 A1 ̸≥ P2 A2 and P1 R1 ̸≥ P2 R2 .
4/3 1
2 −1
1 0
0 0
5. The group inverse analogue This section deals with the group inverse analogue of a few earlier results, and is meant for only square singular matrices. So, we assume all our matrices in this section are square matrices. Analogous to the definition of a proper nonnegative splitting, let us now introduce another definition for square singular matrices using the group inverse for studying the iterative scheme x(i+1) = U # Vx(i) + U # b.
D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
143
Definition 5.1. A decomposition A = U − V of A ∈ Rn×n is called a proper G-nonnegative splitting if it is a proper splitting such that U # exists and U # V ≥ O. Recently, Mishra [19] has shown the following properties of a proper splitting using the group inverse of A. For completeness, we have reproduced the proof here. Theorem 5.2 (Theorem 3.4.1, [19]). Let A = U − V be a proper splitting of A ∈ Rn×n . Suppose that A# exists. Then (a) (b) (c) (d) (e)
U # exists. AA# = UU # ; A# A = U # U. A = U (I − U # V ). I − U # V is invertible. A# = (I − U # V )−1 U # .
Proof. (a) Since A = U − V is a proper splitting, so R(A) = R(U ) and N (A) = N (U ). Since A# exists, R(A) and N (A) are complementary subspaces and so are R(U ) and N (U ). Hence U # exists. (b) Since R(A) = R(U ) = R(A# ) and N (A) = N (U ) = N (U # ), it follows that PR(A),N (A) = PR(U ),N (U ) . Therefore AA# = UU # ; # A A = U # U. (c) Since R(A) = R(U ), it follows that R(V ) ⊆ R(A). Hence UU # V = V and then we have A = U − V = U − UU # V = U (I − U # V ). (d) Let (I − U # V )x = 0. Then x = U # Vx ∈ R(U # ) = R(U ) = R(A) and Ux = UU # Vx = Vx. So 0 = (U − V )x = Ax. We then have x = 0 since x ∈ R(A) ∩ N (A) which in turn implies the fact that I − U # V is invertible. (e) Set X = (I − U # V )−1 U # . Let x ∈ R(A). So x = A# Ax = U # Ux. Then XAx = (I − U # V )−1 U # (U − V )x = (I − U # V )−1 (U # Ux − U # Vx) = (I − U # V )−1 (x − U # Vx) = (I − U # V )−1 (I − U # V )x = x. Again, suppose that y ∈ N (A) = N (U ) = N (U # ). Then Xy = (I − U # V )−1 U # y = 0. This shows that A# = X = (I − U # V )−1 U # . A convergence result for the proper G-nonnegative splitting is shown next. The idea of this proof is very close to the idea of the proof of Lemma 3.4. ρ(A# U )−1
Lemma 5.3. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n and A# U ≥ O. Then ρ(U # V ) = ρ(A# U ) < 1. Proof. From the definition of a proper G-nonnegative splitting, we have A = U − V is a proper splitting with the conditions U # exists and U # V ≥ O. Then by Theorem 2.3, there exists a nonnegative vector x (x ̸= 0) such that U # Vx = ρ(U # V )x. Hence x ∈ R(U # ) = R(U ) = R(A) so that U # Ux = x. Since U # exists, so A# exists and A# = (I − U # V )−1 U # by Theorem 5.2(a) and (d). So A# U = (I − U # V )−1 U # U. Then
(I − U # V )−1 U # Ux = (I − U # V )−1 x =
1 1 − ρ(U # V )
x = A# Ux. ρ(A# U )−1
This implies 1−ρ(1U # V ) ≥ 0 and is an eigenvalue of A# U. Hence 0 ≤ 1−ρ(1U # V ) ≤ ρ(A# U ) i.e., ρ(U # V ) ≤ ρ(A# U ) . Again, the condition A# U ≥ O yields existence of a nonnegative vector y (y ̸= 0) such that A# Uy = ρ(A# U )y. Then y ∈ R(A) = R(U ). Therefore (I − U # V )−1 U # Uy = (I − U # V )−1 y = ρ(A# U )y. So, we have ρ(A#1 U )y = y − U # Vy i.e., U # Vy =
ρ(U V ) ≥ #
ρ(A# U )−1 . ρ(A# U )
Hence ρ(U V ) = #
ρ(A# U )−1 ρ(A# U )
< 1.
ρ(A# U )−1 y. ρ(A# U )
Thus
From the above proof it is clear that the proofs for all the remaining results can be done by using the similar idea as in case of the Moore–Penrose inverse and using the properties of group inverse of a matrix. Hence we present below a few other main results for square singular matrices without their proof. Lemma 5.4. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n . Then A# V ≥ O if and only if ρ(U # V ) = ρ(A# V ) 1+ρ(A# V )
< 1.
Theorem 5.5. Let A = U − V be a proper G-nonnegative splitting of A ∈ Rn×n . Then the following are equivalent:
ρ(U # V ) < 1; A# U − A# A ≥ O; A# U ≥ A# A; A# V ≥ U # V ; A# V ≥ O; ρ(A# V ) (f) ρ(U # V ) = 1+ρ(A# V ) .
(a) (b) (c) (d) (e)
With these results, let us move to the theory of double splittings for square singular matrices. At first, the definition of double proper G-nonnegative splitting is introduced below for index one matrices in connection with the iterative scheme xi+1 = P # Rxi + P # Sxi−1 + P # b.
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D. Mishra / Computers and Mathematics with Applications 67 (2014) 136–144
Definition 5.6. A double decomposition A = P − R − S of A ∈ Rn×n is called double proper G-nonnegative splitting if A# exists, R(A) = R(P ), N (A) = N (P ), P # R ≥ O and P # S ≥ O. Note that the condition A# exists, R(A) = R(P ) and N (A) = N (P ) imply P # exists. A convergence theorem for the double proper G-nonnegative splittings is obtained below. Theorem 5.7. If A = P − R − S is a double proper G-nonnegative splitting with A# P ≥ O, then ρ(W ) < 1. Next one is the comparison result for two square singular linear systems. Theorem 5.8. Let A1 and A2 be two square singular matrices having the same null space. Suppose that A1 = P1 − R1 − S1 and A2 = P2 − R2 − S2 be their double proper G-nonnegative splittings, A#1 P1 ≥ O and A#2 P2 ≥ O. If P1# A1 ≥ P2# A2 and P1# R1 ≥ P2# R2 , then ρ(W1 ) ≤ ρ(W2 ) < 1. 6. Conclusions Many methods have been suggested in the literature to solve the square singular linear systems (1.1). The iterative methods (i.e., matrix splitting methods) and semi-iterative methods are among them. Recently, much attention is drawn to the Krylov subspace methods. It is known that the solution to a nonsingular linear system Ax = b lies in the Krylov space of (A, b)1 whose dimension is the degree of the minimal polynomial of A. However, the Krylov methods can fail if the matrix A is singular, and in some cases even if the linear system does have a solution, it may not lie in a Krylov space. Therefore in these cases, the theory of splittings may be useful. But the existence theorem of a Krylov solution for some square singular linear system says that a linear system Ax = b has a Krylov solution if and only if b ∈ R(Ak ), where k is the index of A. Another well-known result which relates the Krylov solution and the generalized inverse is described next. Let m be the degree of the minimal polynomial for A. If b ∈ R(Ak ), then the linear system Ax = b has a unique Krylov solution x = AD b ∈ Km−k (A, b). Otherwise, Ax = b does not have a solution in the Krylov space Kn (A, b), where n is the order of the matrix A. From the above results, it is clear that computation of AD b (or AD ) is important. This can be done by considering the iterative scheme xi+1 = U D Vxi + U D b, i = 0, 1, 2, . . . (or X i+1 = U D VX i + U D , i = 0, 1, 2, . . . , where X denotes a matrix). The convergence of the above iterative scheme will go in a very similar way as followed in this article and will be studied in future. Acknowledgements The author thanks the anonymous referees for their valuable suggestions which improved the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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1 K (A, b) = span{b, Ab, A2 b, . . . , As−1 b}, where s denotes the dimension. s