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A note on the generalized Bott–Duffin inverse
Applied Mathematics Letters 20 (2007) 746–750 www.elsevier.com/locate/aml
A note on the generalized Bott–Duffin inverse Bin Deng, GuoLiang Chen ∗ Department of Mathematics, East China Normal University, Shanghai, 200062, PR China Received 5 November 2003; received in revised form 29 July 2005; accepted 28 June 2006
Abstract By generalizing subspaces T, T ⊥ found in [G. Chen, G. Liu, Y. Xue, Perturbation analysis of the generalized Bott–Duffin inverse of L-zero matrices, Linear Multilinear Algebra 51 (2003) 11–20] to subspaces T, S ⊂ C n , for which T ⊕ S = C n , we (+) get that if AT ∩ S = 0, A T,S = PT,S (A PT,S + PS,T )+ is (PT,S A PT,S )+ , then S = T ⊥ , that is to say A is a T -zero matrix (i.e. AT ∩ T ⊥ = 0). c 2006 Elsevier Ltd. All rights reserved.
Keywords: Generalized Bott–Duffin inverse; L-zero matrix
1. Introduction In their seminal paper [2], Bott and Duffin introduced and widely used an important tool called the “constrained inverse” of the matrix. This inverse is called the Bott–Duffin inverse. Ben Israel and Greville in [1] mentioned many properties and applications. Later, Chen in his paper [5] defined the generalized Bott–Duffin inverse and gave some properties and applications. G. Chen, G. Liu, Y. Xue in papers [3,4,6] defined L-zero matrices in order to simplify the expression for the generalized Bott–Duffin inverse. We adopt the same notation for the generalized inverses of matrices as that in [1]. In this work, we will discuss a general case of the generalized Bott–Duffin inverse. Consider the special inverse (+)
A T,S = PT,S (A PT,S + PS,T )+
(∗)
of a matrix A ∈ C n×n , where T and S are subspaces of C n such that T ⊕ S = C n . Throughout the work, A ∈ C n×n , T and S denote subspaces of C n with T ⊕ S = C n and dim (T ) = r . Let {ε1 , ε2 , . . . , εr } be an orthonormal basis of T , and {εr +1 , εr +2 , . . . , εn } be an orthonormal basis of S. Let W = (ε1 , ε2 , . . . , εn ) and define matrix B by B11 B12 B11 B12 AW = W B = W = (ε1 , ε2 , . . . , εn ) B21 B22 B21 B22 with B11 ∈ C r ×r . ∗ Corresponding author.
E-mail addresses: [email protected] (B. Deng), [email protected] (G. Chen). c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.06.018
747
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
From the orthogonality, we get Ir C ∗ W W = C ∗ In−r ε∗
(1)
1 ∗
ε2 with C = . (εr +1 , εr +2 , . . . , εn ) ∈ C r ×(n−r ) . . . εr∗
From [7] it follows that (W W ) ∗
−1
Ir = C∗
C
−1
(I − CC ∗ )−1 = C ∗ (CC ∗ − I )−1
In−r
C(C ∗ C − I )−1 . (I − C ∗ C)−1
(2)
2. Main results Lemma 1 ([1]). (1) PT,S A = A iff R(A) ⊂ T (2) A PT,S = A iff N (A) ⊃ S. (+)
Lemma 2 ([3]). Let A ∈ C n×n , T be a subspace of C n with orthonormal basis {ε1 , ε2 , . . . , εr }, and A T PT (A PT + PT ⊥ )+ . Then the following statements are equivalent:
=
(1) AT ∩ T ⊥ = {0}, (+) (2) A T = (PT A PT )+ , (3) rank(AU ) = rank(U ∗ AU ), where U = (ε1 , ε2 , . . . , εr ). (+)
Theorem 1. If AT ∩ S = 0 and A T,S = (PT,S A PT,S )+ , then S = T ⊥ (i.e. the case of the generalized Bott–Duffin inverse). B12 11 Proof. From PT,S W = (ε1 , ε2 , . . . , εr , 0, . . . , 0) and AW = W BB21 B22 , it follows that 0 W −1 0 0 0 PS,T = (0, . . . , 0, εr +1 , εr +2 , . . . , εn )W −1 = W W −1 0 In−r B11 B12 I 0 0 0 A PT,S + PS,T = W W −1 + W W −1 W −1 W r B21 B22 0 0 0 In−r B11 0 = W W −1 B21 In−r I 0 B11 B12 I 0 W −1 W PT,S A PT,S = W r W −1 W r W −1 0 0 B21 B22 0 0 B11 0 W −1 . = W 0 0 PT,S = (ε1 , ε2 , . . . , εr , 0, . . . , 0)W −1 = W
Ir 0
From AT ∩ S = 0, it follows that PS,T A PT,S = 0. And PS,T + PT,S = I ; thus (I − PT,S )A PT,S = 0, i.e. PT,S A PT,S = A PT,S . Hence B21 = 0. Let E = (A PT,S + PS,T ). From PT,S (A PT,S + PS,T )+ = (PT,S A PT,S )+ = (A PT,S )+
(3)
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B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
and the definition of the Moore–Penrose inverse, we can get PT,S E + A PT,S E + = PT,S E +
(4)
+
(5)
A PT,S E A PT,S = A PT,S (A PT,S E ) = A PT,S E + ∗
+
(6)
(PT,S E A PT,S ) = PT,S E A PT,S . +
∗
+
(7)
Premultiplying (7) by A and from (5), ∗ ∗ A PT,S = A PT,S A∗ (E + )∗ PT,S .
(8)
Postmultiplying (8) by A∗ and from (5), ∗ A PT,S A∗ = A PT,S A∗ ,
i.e. ∗ ∗ B11 0 0 B11 B12 ∗ −1 B11 ∗ ∗ −1 B11 (W W ) W =W W (W W ) ∗ 0 0 B12 In−r 0 In−r 0 ∗ B11 0 (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 0 × ∗ ∗ ∗ −1 ∗ −1 0 0 B12 In−r C (CC − I ) (I − C C) ∗ B11 B12 (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 0 = 0 In−r 0 0 C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 ∗ −1 ∗ ∗ −1 ∗ B11 (I − CC ) B11 + B11 C(C C − I ) B12 B11 C(C ∗ C − I )−1 × 0 0 ∗ ∗ B11 (I − CC ∗ )−1 B11 + B12 C ∗ (CC ∗ − I )−1 B11 0 = . ∗ ∗ −1 ∗ C (CC − I ) B11 0
0 W∗ 0
(9)
It is easy to verify C ∗ (CC ∗ − I )−1 = (C ∗ C − I )−1 C ∗ , from (9); thus B11 C = 0. From (5) and (6), we can get A PT,S = (A PT,S E + )∗ A PT,S = (E + )∗ (A PT,S )∗ A PT,S .
(10)
Premultiplying (10) by E E ∗ , E E ∗ A PT,S = E E ∗ (E + )∗ (A PT,S )∗ A PT,S = E(E + E)∗ (A PT,S )∗ A PT,S = E(A PT,S )∗ A PT,S , i.e. W
0
B11 0
In−r
= W
B11 0
∗ B11 0
0
(W W ) In−r ∗ 0 B11 (W ∗ W )−1 In−r 0 ∗
−1
B11 0 W W W −1 0 0 0 B11 0 ∗ (W W ) W −1 . 0 0 0 ∗
So 0=W 0 = 0
B11 0
0 In−r
(W W ) ∗
−1
0 ∗ (I − C C)−1 C ∗ B11
and thus C ∗ B11 = 0.
0 0
0 In−r
∗
W W
B11 0
0 W −1 0
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750 + + + + + ∗ ∗ Then B11 C = B11 B11 B11 C = B11 (B11 ) B11 C = 0. + For C B11 = 0 and B11 C = 0, it is easy to verify that + + B11 0 B11 + −1 =W (PT,S A PT,S ) = W W 0 0 0
0 W −1 . 0
749
(11)
Let (A PT,S + PS,T )+ = W
F11 F21
F12 W −1 . F22
Substituting (11) and (12) into (3) yields + I 0 F11 F12 B11 W r W −1 W W −1 = W 0 0 F21 F22 0
(12)
0 W −1 . 0
+ It follows that F11 = B11 and F12 = 0. From (A PT,S + PS,T )(A PT,S + PS,T )+ (A PT,S + PS,T ) = (A PT,S + PS,T ), + B11 0 B11 0 B11 0 B11 0 = 0 In−r 0 In−r 0 In−r F21 F22 + B11 B11 B11 0 = . F21 B11 F22
Hence F22 = In−r . From ((A PT,S + PS,T )(A PT,S + PS,T )+ )∗ = (A PT,S + PS,T )(A PT,S + PS,T )+ , ∗ + + B11 B11 0 B11 B11 0 W W −1 = W W −1 F21 In−r F21 In−r + ∗ B11 B11 F21 0 In−r + (B11 B11 + C F21 )(I − CC ∗ )−1 + CC ∗ (CC ∗ − I )−1 C F21 C(C ∗ C − I )−1 + C(I − C ∗ C)−1 = F21 (I − CC ∗ )−1 + C ∗ (CC ∗ − I )−1 F21 C(C ∗ C − I )−1 + (I − C ∗ C)−1 and then F21 (I − CC ∗ )−1 + C ∗ (CC ∗ − I )−1 = 0, so F21 = C ∗ . For ((A PT,S + PS,T )+ (A PT,S + PS,T ))∗ = (A PT,S + PS,T )+ (A PT,S + PS,T ), + ∗ + B11 B11 0 B11 B11 0 W W −1 = W W −1 0 In−r 0 In−r + + Ir C (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 B11 0 B11 B11 0 = C ∗ In−r 0 In−r 0 In−r C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 + + B11 B11 (I − CC ∗ )−1 + CC ∗ (CC ∗ − I )−1 B11 B11 C(C ∗ C − I )−1 + C(I − C ∗ C)−1 = C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 and thus C ∗ (CC ∗ − I )−1 = 0, so it follows that C = 0. (+) (+) From the definition of C, we can easily get S = T ⊥ (i.e. A T,S = A T ).
Acknowledgements This work was supported by Shanghai Priority Academic Discipline (04JC14031) and NNSF, PR China (10371044). And the authors are very grateful for the suggestions that led to considerable improvement of an earlier version of this work. References [1] A. Ben Israel, T.N.E. Greville, Generalized Inverse: Theory and Applications, 2nd ed., Springer-Verlag, New York, 2003.
750
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
[2] R. Bott, R.J. Duffin, On the algebra of network, Trans. Amer. Math. Soc. 74 (1953) 99–109. [3] G. Chen, G. Liu, Y. Xue, Perturbation analysis of the generalized Bott–Duffin inverse of L-zero matrices, Linear Multilinear Algebra 51 (2003) 11–20. [4] G. Chen, G. Liu, Y. Xue, Perturbation theory for the generalized Bott–Duffin inverse and its applications, Appl. Math. Comput. 129 (2002) 145–155. [5] Y. Chen, The generalized Bott–Duffin inverse and its applications, Linear Algebra Appl. 134 (1990) 71–91. [6] Y. Xue, G. Chen, The expression of the generalized Bott–Duffin inverse and its perturbation theory, Appl. Math. Comput. 132 (2002) 437–444. [7] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (Cambridgeshire), New York, 1985.
A note on the generalized Bott–Duffin inverse Bin Deng, GuoLiang Chen ∗ Department of Mathematics, East China Normal University, Shanghai, 200062, PR China Received 5 November 2003; received in revised form 29 July 2005; accepted 28 June 2006
Abstract By generalizing subspaces T, T ⊥ found in [G. Chen, G. Liu, Y. Xue, Perturbation analysis of the generalized Bott–Duffin inverse of L-zero matrices, Linear Multilinear Algebra 51 (2003) 11–20] to subspaces T, S ⊂ C n , for which T ⊕ S = C n , we (+) get that if AT ∩ S = 0, A T,S = PT,S (A PT,S + PS,T )+ is (PT,S A PT,S )+ , then S = T ⊥ , that is to say A is a T -zero matrix (i.e. AT ∩ T ⊥ = 0). c 2006 Elsevier Ltd. All rights reserved.
Keywords: Generalized Bott–Duffin inverse; L-zero matrix
1. Introduction In their seminal paper [2], Bott and Duffin introduced and widely used an important tool called the “constrained inverse” of the matrix. This inverse is called the Bott–Duffin inverse. Ben Israel and Greville in [1] mentioned many properties and applications. Later, Chen in his paper [5] defined the generalized Bott–Duffin inverse and gave some properties and applications. G. Chen, G. Liu, Y. Xue in papers [3,4,6] defined L-zero matrices in order to simplify the expression for the generalized Bott–Duffin inverse. We adopt the same notation for the generalized inverses of matrices as that in [1]. In this work, we will discuss a general case of the generalized Bott–Duffin inverse. Consider the special inverse (+)
A T,S = PT,S (A PT,S + PS,T )+
(∗)
of a matrix A ∈ C n×n , where T and S are subspaces of C n such that T ⊕ S = C n . Throughout the work, A ∈ C n×n , T and S denote subspaces of C n with T ⊕ S = C n and dim (T ) = r . Let {ε1 , ε2 , . . . , εr } be an orthonormal basis of T , and {εr +1 , εr +2 , . . . , εn } be an orthonormal basis of S. Let W = (ε1 , ε2 , . . . , εn ) and define matrix B by B11 B12 B11 B12 AW = W B = W = (ε1 , ε2 , . . . , εn ) B21 B22 B21 B22 with B11 ∈ C r ×r . ∗ Corresponding author.
E-mail addresses: [email protected] (B. Deng), [email protected] (G. Chen). c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.06.018
747
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
From the orthogonality, we get Ir C ∗ W W = C ∗ In−r ε∗
(1)
1 ∗
ε2 with C = . (εr +1 , εr +2 , . . . , εn ) ∈ C r ×(n−r ) . . . εr∗
From [7] it follows that (W W ) ∗
−1
Ir = C∗
C
−1
(I − CC ∗ )−1 = C ∗ (CC ∗ − I )−1
In−r
C(C ∗ C − I )−1 . (I − C ∗ C)−1
(2)
2. Main results Lemma 1 ([1]). (1) PT,S A = A iff R(A) ⊂ T (2) A PT,S = A iff N (A) ⊃ S. (+)
Lemma 2 ([3]). Let A ∈ C n×n , T be a subspace of C n with orthonormal basis {ε1 , ε2 , . . . , εr }, and A T PT (A PT + PT ⊥ )+ . Then the following statements are equivalent:
=
(1) AT ∩ T ⊥ = {0}, (+) (2) A T = (PT A PT )+ , (3) rank(AU ) = rank(U ∗ AU ), where U = (ε1 , ε2 , . . . , εr ). (+)
Theorem 1. If AT ∩ S = 0 and A T,S = (PT,S A PT,S )+ , then S = T ⊥ (i.e. the case of the generalized Bott–Duffin inverse). B12 11 Proof. From PT,S W = (ε1 , ε2 , . . . , εr , 0, . . . , 0) and AW = W BB21 B22 , it follows that 0 W −1 0 0 0 PS,T = (0, . . . , 0, εr +1 , εr +2 , . . . , εn )W −1 = W W −1 0 In−r B11 B12 I 0 0 0 A PT,S + PS,T = W W −1 + W W −1 W −1 W r B21 B22 0 0 0 In−r B11 0 = W W −1 B21 In−r I 0 B11 B12 I 0 W −1 W PT,S A PT,S = W r W −1 W r W −1 0 0 B21 B22 0 0 B11 0 W −1 . = W 0 0 PT,S = (ε1 , ε2 , . . . , εr , 0, . . . , 0)W −1 = W
Ir 0
From AT ∩ S = 0, it follows that PS,T A PT,S = 0. And PS,T + PT,S = I ; thus (I − PT,S )A PT,S = 0, i.e. PT,S A PT,S = A PT,S . Hence B21 = 0. Let E = (A PT,S + PS,T ). From PT,S (A PT,S + PS,T )+ = (PT,S A PT,S )+ = (A PT,S )+
(3)
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B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
and the definition of the Moore–Penrose inverse, we can get PT,S E + A PT,S E + = PT,S E +
(4)
+
(5)
A PT,S E A PT,S = A PT,S (A PT,S E ) = A PT,S E + ∗
+
(6)
(PT,S E A PT,S ) = PT,S E A PT,S . +
∗
+
(7)
Premultiplying (7) by A and from (5), ∗ ∗ A PT,S = A PT,S A∗ (E + )∗ PT,S .
(8)
Postmultiplying (8) by A∗ and from (5), ∗ A PT,S A∗ = A PT,S A∗ ,
i.e. ∗ ∗ B11 0 0 B11 B12 ∗ −1 B11 ∗ ∗ −1 B11 (W W ) W =W W (W W ) ∗ 0 0 B12 In−r 0 In−r 0 ∗ B11 0 (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 0 × ∗ ∗ ∗ −1 ∗ −1 0 0 B12 In−r C (CC − I ) (I − C C) ∗ B11 B12 (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 0 = 0 In−r 0 0 C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 ∗ −1 ∗ ∗ −1 ∗ B11 (I − CC ) B11 + B11 C(C C − I ) B12 B11 C(C ∗ C − I )−1 × 0 0 ∗ ∗ B11 (I − CC ∗ )−1 B11 + B12 C ∗ (CC ∗ − I )−1 B11 0 = . ∗ ∗ −1 ∗ C (CC − I ) B11 0
0 W∗ 0
(9)
It is easy to verify C ∗ (CC ∗ − I )−1 = (C ∗ C − I )−1 C ∗ , from (9); thus B11 C = 0. From (5) and (6), we can get A PT,S = (A PT,S E + )∗ A PT,S = (E + )∗ (A PT,S )∗ A PT,S .
(10)
Premultiplying (10) by E E ∗ , E E ∗ A PT,S = E E ∗ (E + )∗ (A PT,S )∗ A PT,S = E(E + E)∗ (A PT,S )∗ A PT,S = E(A PT,S )∗ A PT,S , i.e. W
0
B11 0
In−r
= W
B11 0
∗ B11 0
0
(W W ) In−r ∗ 0 B11 (W ∗ W )−1 In−r 0 ∗
−1
B11 0 W W W −1 0 0 0 B11 0 ∗ (W W ) W −1 . 0 0 0 ∗
So 0=W 0 = 0
B11 0
0 In−r
(W W ) ∗
−1
0 ∗ (I − C C)−1 C ∗ B11
and thus C ∗ B11 = 0.
0 0
0 In−r
∗
W W
B11 0
0 W −1 0
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750 + + + + + ∗ ∗ Then B11 C = B11 B11 B11 C = B11 (B11 ) B11 C = 0. + For C B11 = 0 and B11 C = 0, it is easy to verify that + + B11 0 B11 + −1 =W (PT,S A PT,S ) = W W 0 0 0
0 W −1 . 0
749
(11)
Let (A PT,S + PS,T )+ = W
F11 F21
F12 W −1 . F22
Substituting (11) and (12) into (3) yields + I 0 F11 F12 B11 W r W −1 W W −1 = W 0 0 F21 F22 0
(12)
0 W −1 . 0
+ It follows that F11 = B11 and F12 = 0. From (A PT,S + PS,T )(A PT,S + PS,T )+ (A PT,S + PS,T ) = (A PT,S + PS,T ), + B11 0 B11 0 B11 0 B11 0 = 0 In−r 0 In−r 0 In−r F21 F22 + B11 B11 B11 0 = . F21 B11 F22
Hence F22 = In−r . From ((A PT,S + PS,T )(A PT,S + PS,T )+ )∗ = (A PT,S + PS,T )(A PT,S + PS,T )+ , ∗ + + B11 B11 0 B11 B11 0 W W −1 = W W −1 F21 In−r F21 In−r + ∗ B11 B11 F21 0 In−r + (B11 B11 + C F21 )(I − CC ∗ )−1 + CC ∗ (CC ∗ − I )−1 C F21 C(C ∗ C − I )−1 + C(I − C ∗ C)−1 = F21 (I − CC ∗ )−1 + C ∗ (CC ∗ − I )−1 F21 C(C ∗ C − I )−1 + (I − C ∗ C)−1 and then F21 (I − CC ∗ )−1 + C ∗ (CC ∗ − I )−1 = 0, so F21 = C ∗ . For ((A PT,S + PS,T )+ (A PT,S + PS,T ))∗ = (A PT,S + PS,T )+ (A PT,S + PS,T ), + ∗ + B11 B11 0 B11 B11 0 W W −1 = W W −1 0 In−r 0 In−r + + Ir C (I − CC ∗ )−1 C(C ∗ C − I )−1 B11 B11 0 B11 B11 0 = C ∗ In−r 0 In−r 0 In−r C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 + + B11 B11 (I − CC ∗ )−1 + CC ∗ (CC ∗ − I )−1 B11 B11 C(C ∗ C − I )−1 + C(I − C ∗ C)−1 = C ∗ (CC ∗ − I )−1 (I − C ∗ C)−1 and thus C ∗ (CC ∗ − I )−1 = 0, so it follows that C = 0. (+) (+) From the definition of C, we can easily get S = T ⊥ (i.e. A T,S = A T ).
Acknowledgements This work was supported by Shanghai Priority Academic Discipline (04JC14031) and NNSF, PR China (10371044). And the authors are very grateful for the suggestions that led to considerable improvement of an earlier version of this work. References [1] A. Ben Israel, T.N.E. Greville, Generalized Inverse: Theory and Applications, 2nd ed., Springer-Verlag, New York, 2003.
750
B. Deng, G. Chen / Applied Mathematics Letters 20 (2007) 746–750
[2] R. Bott, R.J. Duffin, On the algebra of network, Trans. Amer. Math. Soc. 74 (1953) 99–109. [3] G. Chen, G. Liu, Y. Xue, Perturbation analysis of the generalized Bott–Duffin inverse of L-zero matrices, Linear Multilinear Algebra 51 (2003) 11–20. [4] G. Chen, G. Liu, Y. Xue, Perturbation theory for the generalized Bott–Duffin inverse and its applications, Appl. Math. Comput. 129 (2002) 145–155. [5] Y. Chen, The generalized Bott–Duffin inverse and its applications, Linear Algebra Appl. 134 (1990) 71–91. [6] Y. Xue, G. Chen, The expression of the generalized Bott–Duffin inverse and its perturbation theory, Appl. Math. Comput. 132 (2002) 437–444. [7] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (Cambridgeshire), New York, 1985.