Moore-Penrose Inverses of Perturbed Operators on Hilbert Spaces
Moore-Penrose Inverses of Perturbed Operators on Hilbert Spaces Shani Jose
Joint work with: Dr. K. C. Sivakumar Indian Institute of Technology Madras, Chennai
Prairie Analysis Seminar
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
1 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse The best approximate solution (least square solution of minimum norm): x = A† b
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse The best approximate solution (least square solution of minimum norm): x = A† b
A† – Moore-Penrose inverse Computational techniques Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Moore-Penrose Inverse Let A ∈ B(H1 , H2 ). The Moore-Penrose inverse of A is the unique operator A† ∈ B(H2 , H1 ) (if it exists) satisfying 1
AA† A = A
2
A† AA† = A†
3
(AA† )∗ = AA†
4
(A† A)∗ = A† A If A is nonsingular, then A† = A−1 . A† exists if and only if R(A) is closed. Exists for all matrices.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
3 / 29
Remarks Equivalent Definition for A† (i) A† Ax = x, ∀x ∈ R(A∗ ) (ii) A† y = 0, ∀y ∈ N(A∗ ).
Properties of A† R(A† ) = R(A∗ ),
N(A† ) = N(A∗ ),
AA† = PR(A) ,
A† A = PR(A∗ )
PL : – the orthogonal projection onto the space L.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
4 / 29
Motivation There are many physical models where successive computation of matrix inverses must be performed. The amount of computation increases rapidly as the order of the matrix increases. Most cases, the matrix that we deal with may not be the exact matrix representing the data. Truncation and round off errors give rise to perturbations. If we have a nonsingular matrix and its inverse, and suppose that some of the entries of the matrix are altered, then it would be desirable to have an efficient computational method for finding the inverse of the new matrix from the known inverse without computing the inverse of the new matrix, afresh. =⇒ Sherman-Morrison-Woodbury (SMW) formula
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
5 / 29
Sherman-Morrison-Woodbury formula Let A be an n × n and G be an r × r nonsingular matrices with r ≤ n. Also, let V1 and V2 be n × r matrices such that (G −1 + V2 ∗ A−1 V1 )−1 exists. Then (A + V1 GV2∗ )−1 = A−1 − A−1 V1 (G −1 + V2 ∗ A−1 V1 )−1 V2 ∗ A−1 An explicit formula for the inverse of matrices of the form A + V1 G V2 ∗ . Useful: I I I
Situations where r is much smaller than n When A has certain structural properties Computing A−1 V1 (G −1 + V2 ∗ A−1 V1 )−1 V2 ∗ A−1 is ‘easier’ than inverting a general perturbed n × n matrix
Applications: Appears in Statistics, networks, structural analysis, asymptotic analysis, optimization and PDEs1 . 1
W. W. Hager. Updating the inverse of a matrix, SIAM Review, 1989. Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
6 / 29
Objective: Extend SMW formula to operators on Hilbert spaces. Obtain conditions for the nonnegativity of Moore-Penrose inverse of perturbed operators and matrices.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
7 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then • M is a rank-one perturbation of A.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then • M is a rank-one perturbation of A. • M ∈ B(H1 , H2 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Mutually Exclusive Cases3,4
For λ = 1 + A† b, c , (i) b ∈ R(A), c ∈ R(A∗ ) and λ = 0. (ii) b ∈ R(A), c ∈ R(A∗ ) and λ 6= 0. (iii) b ∈ / R(A), c ∈ / R(A∗ ). (iv) b ∈ R(A), c ∈ / R(A∗ ). (v) b ∈ / R(A), c ∈ R(A∗ ). Note: The cases are collectively exhaustive. 3
C. D. Meyer, Generalized inversion of modified matrices. SIAM J. Appl. Math., 1973.
4
J. K. Baksalary, O. M. Baksalary, G. Trenkler, A revisitation formulae for the Moore-Penrose inverse of modified matrices. Lin.
Alg. Appl. 2003.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
9 / 29
Notation
d = A† b,
e = (A† )∗ c,
δ = hd, di , η = he, ei , µ = |λ|2 + δψ,
f = PR(A) ⊥ b, φ = hf , f i ,
ν = |λ|2 + ηφ,
g = PR(A∗ ) ⊥ c ψ = hg , g i
¯ + δg ∈ H1 p = λd
q = λe + ηf ∈ H2 where λ = 1 + hd, ci = 1 + hb, ei .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
10 / 29
Case (i) R(M) is closed. Set D : H1 → H1 , E : H2 → H2 by D(x) = hx, di d and E (x) = hx, ei e. Then M † = A† − δ −1 DA† − η −1 A† E + δ −1 η −1 DA† E where d = A† b, e = (A† )∗ c, δ = hd, di , η = he, ei .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
11 / 29
Case (i) R(M) is closed. Set D : H1 → H1 , E : H2 → H2 by D(x) = hx, di d and E (x) = hx, ei e. Then M † = A† − δ −1 DA† − η −1 A† E + δ −1 η −1 DA† E where d = A† b, e = (A† )∗ c, δ = hd, di , η = he, ei .
Case (ii) Assume R(M) is closed. Then M † = A† − λ−1 A† BA† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
11 / 29
Case (iii) Assume that R(M) closed. Set B1 : H2 → H2 by B1 (x) = hx, f i b, G1 : H1 → H1 by G1 (x) = hx, ci g , G2 : H2 → H1 by G2 (x) = hx, f i g . Then M † = A† − φ−1 A† B1 − ψ −1 G1 A† + λφ−1 ψ −1 G2 where f = PR(A) ⊥ b, g = PR(A∗ ) ⊥ c, φ = hf , f i ,
Shani (2013)
Generalized Inverses
ψ = hg , g i .
Prairie Analysis Seminar
12 / 29
Case (iii) Assume that R(M) closed. Set B1 : H2 → H2 by B1 (x) = hx, f i b, G1 : H1 → H1 by G1 (x) = hx, ci g , G2 : H2 → H1 by G2 (x) = hx, f i g . Then M † = A† − φ−1 A† B1 − ψ −1 G1 A† + λφ−1 ψ −1 G2 where f = PR(A) ⊥ b, g = PR(A∗ ) ⊥ c, φ = hf , f i ,
ψ = hg , g i .
Case (iv) Let R(M) be closed. Set G3 : H1 → H1 by G3 (x) = hx, di g . Let D and G1 be defined as in the previous theorems. Then ¯ † BA† ) M † = A† − µ−1 (ψDA† + δG1 A† − λG3 A† + λA where µ = |λ|2 + δψ.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
12 / 29
Case (v) ˜ Set B(x) = hx, f i b and E˜ (x) = hx, f i e. Let E be as before. Let R(M) be closed. Then ˜ − λA† E˜ + λA ¯ † BA† ) M † = A† − ν −1 (φA† E + ηA† B where ν = |λ|2 + ηφ.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
13 / 29
Case (v) ˜ Set B(x) = hx, f i b and E˜ (x) = hx, f i e. Let E be as before. Let R(M) be closed. Then ˜ − λA† E˜ + λA ¯ † BA† ) M † = A† − ν −1 (φA† E + ηA† B where ν = |λ|2 + ηφ. The above 5 cases are generalized to obtain formulae for (A + V1 GV2∗ )† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
13 / 29
Example Let H1 = H2 = B(l 2 ) and A ∈ B(l 2 ) be the right shift operator. Then • A† = A∗ - the left shift operator. • B(x) = hx, ci b, for fixed nonzero vectors b, c ∈ l2 and M = A + B. • R(M) is closed5 if kbk kck < 1. Take b = e1 and c = e3 where ei ∈ l 2 is the vector containing i th element 1 and all other elements zero. b∈ / R(A) and c ∈ R(A∗ ) M † (x) = A† x − 21 (x4 − x1 )e3 , where x = (x1 , x2 , x3 , . . .) ∈ l 2 . 5
O. Christensen, Operators with closed range, pseudo-Inverses, and perturbation of frames for a subspace. Canad. Math.
Bull.,1999.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
14 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2 Then ˆ is closed. X R(Ω)
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2 Then ˆ is closed. X R(Ω) ˆ † = PR(A† V )⊥ A† P † ∗ ⊥ X Ω R(A V2 ) 1
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ )
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ ) • Ω = A + V1 GV2∗ have closed range space.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ ) • Ω = A + V1 GV2∗ have closed range space. Then, Ω† = A† − A† V1 (G † + V2∗ A† V1 )† V2∗ A† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ). • Ω = A + (U1 + W1 )G (U2 + W2 )∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ). • Ω = A + (U1 + W1 )G (U2 + W2 )∗ with R(Ω) closed. Then ∗
∗
Ω† = A† − W2 † U2 ∗ A† − A† U1 W1 † + W2 † (G † + U2∗ A† U1 )W1 † .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L. • Ω = A + V1 (U2 + W2 )∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L. • Ω = A + V1 (U2 + W2 )∗ with R(Ω) closed. Then Ω† = A† − (A† V1 )K −1 (A† V1 )∗ A† − W2 K −1 U2 ∗ A† + W2 K −1 L(A† V1 )∗ A† − (A† V1 )K −1 L∗ U2 ∗ A† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 )
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
• Ω = A + (U1 + W1 )V2 ∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
• Ω = A + (U1 + W1 )V2 ∗ with R(Ω) closed. Then ∗
∗
Ω† = A† − A† (A† V2 )N −1 (A† V2 )∗ − A† U1 N −1 W1 ∗ ∗ ˜ −1 W1 ∗ − A† U1 L ˜∗ N −1 (A† ∗ V2 )∗ . + A† A† V2 LN
Certain applications of these formulae are obtained. Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Example 1
• H = l 2 , A be the right shift operator on H. • X , Y ∈ B(H) be s. t. −X (x1 , x2 , x3 , . . .) = (0, 0, 0, x1 , x2 , x3 , . . .) Y (x1 , x2 , x3 , . . .) = (x1 , x2 , x1 , x2 , x3 , x4 , . . .). • Set Ω = A + XY ∗ . X R(X ) ⊆ R(A), R(Y ) ⊆ R(A∗ ) and −Y ∗ A† X = I – case (i). X Ω† (x1 , x2 , . . .) = (
Shani (2013)
x2 − x4 x3 − x5 , , 0, 0, . . .). 2 2
Generalized Inverses
Prairie Analysis Seminar
20 / 29
Example 2 • H = l 2 and A ∈ B(H) be A(x1 , x2 , . . .) = (x1 , 0, x2 , 0, x3 , 0, . . .). • R(A) is a closed subspace of l 2 . and A† (x) = A∗ (x) = (x1 , x3 , x5 , . . .) for all x = (x1 , x2 , x3 , . . .). • Let X (x1 , x2 , . . .) = (x1 , 0, x2 , 0, 0, 0, . . .) and Y (x1 , x2 , . . .) = (x1 , x3 , 0, 0, 0, . . .) for all x ∈ H • Ω = A + XY ∗ . X R(Ω) is closed. X Both X , Y ∈ B(H) and R(X ) ⊆ R(A), R(Y ) ⊆ R(A∗ ) and (I + Y ∗ A† X )(x1 , x2 , x3 , . . .) = (2x1 , x2 , x3 + x2 , x4 , x5 , . . .) is bijective. X Ω† (x) = (A† − A† X (I + Y ∗ A† X )−1 Y ∗ A† )(x) = (
Shani (2013)
Generalized Inverses
x1 , x3 , x5 , x7 , x9 , . . .). 2
Prairie Analysis Seminar
21 / 29
Application • H1 = H2 = l2 and A be the right shift operator defined on l 2 . • Then R(A) is closed and hence it has a bounded Moore-Penrose inverse. • A† x = A∗ x = (x2 , x3 , x4 ...), for every x = (x1 , x2 , x3 , ...) ∈ l2 , the left shift operator on l2 . • A ≥ 0 and A† ≥ 0. • Let b = he2 and c = −e1 with h ∈ / {0, 1}. • Take M = A + B, where B = h hx, −e1 i e2 . • R(M) is closed.
• b ∈ R(A), c ∈ R(A∗ ) and λ = 1 + A† b, c = 1 − h 6= 0. Thus we are in Case (ii). • M † (x) = A† (x) − X M † ≥ 0 iff h ∈ Shani (2013)
h 1−h x2 e1 . (−∞, 21 )\{0}
∪ (1, ∞)
Generalized Inverses
Prairie Analysis Seminar
22 / 29
Star partial order and rank-one perturbation Let A1 , A2 ∈ B(H) with R(A1 ) and R(A2 ) closed. The star partial order relating A1 and A2 is defined as ∗
A1 ≤ A2 if and only if A1 A1 † = A2 A1 † and A1 † A1 = A1 † A2 .
To verify whether a rank-one perturbation preserves the star partial order ∗
Let A1 ≤ A2 . b and c in H and B(x) = hx, ci b. Set M1 = A1 + B and M2 = A2 + B. ∗
Let A1 , A2 ∈ B(H1 , H2 ) such that A1 ≤ A2 . Let B = b ⊗ c where b ∈ H2 , ∗
c ∈ H1 and M1 = A1 + B, M2 = A2 + B. Then M1 ≤ M2 if b ∈ R(A1 ) and c ∈ R(A∗1 ). Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
23 / 29
Remark Rank-one perturbations do not preserve star partial order for cases (iii), (iv) and (v) even in the finite dimensional case. Examples: A1 =
1 0 0 0
and A2 =
1 0 0 1
.
∗
Then we have A1 ≤ A2 . 0 • Take b = c = . Then b ∈ / R(A1 ) and c ∈ / R(A∗1 ) and this 1 1 0 1 0 corresponds to case (iii). Here M1 = and M2 = . 0 1 0 2 ∗
But M1 M1T 6= M2 M1T and so M1 ≤ M2 does not hold.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
24 / 29
−1 0 and c = . Here, b ∈ R(A1 ) and c ∈ / R(A∗1 ), 0 1 and so we are in case (iv). Now, consider the rank-one perturbations M1 = A1 + bc T and M2 = A + bc T . Then, 1 −1 1 −1 M1 = and M2 = . 0 0 0 1
• Take b =
Here again, we have M1 M1T 6= M2 M1T . • By interchanging the roles of b, c and A with c, b and A∗ , respectively, it follows that the result does not hold in case (v).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
25 / 29
Theorem ∗
A1 ≤ A2 . B(x) = hx, ci b for fixed nonzero vectors b, c ∈ H. M1 = A1 + B and M2 = A2 + B. Then any two of the following imply the third: ∗
(a) A1 ≤ A2 ∗
(b) M1 ≤ M2 (c) R(B) ⊆ N(A2 − A1 )∗ and R(B ∗ ) ⊆ N(A2 − A1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
26 / 29
Shani Jose and K. C. Sivakumar, Moore-Penrose inverses of perturbed operators on Hilbert spaces, Combinatorial Matrix Theory and Generalized Inverses of Matrices, Springer, India (2013) 119–131. Shani Jose and K. C. Sivakumar, On Nonnegative Moore-Penrose Inverses of Perturbed Matrices. J. Appl. Math., (2013), Article ID 680975, 7 pages.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
27 / 29
References J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, The Ann. Math. Statist., 21 (1950), 124–127. W. W. Hager, Updating the inverse of a matrix, SIAM Review, 31 (1989), 221–239. K. S. Riedel, A Sherman-Morrison-Woodbury identity for rank augmenting matrices with application to centering, SIAM J. Matrix Anal. Appl., 13 (1992), 659–662. J. K. Baksalary, O. M. Baksalary, and G. Trenkler, A revisitation formulae for the moore-penrose inverse of modified matrices. Linear Algebra and Applications, 372 (2003), 207–224.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
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Thank you
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
29 / 29
Joint work with: Dr. K. C. Sivakumar Indian Institute of Technology Madras, Chennai
Prairie Analysis Seminar
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
1 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse The best approximate solution (least square solution of minimum norm): x = A† b
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Inverse problem Solve for x given b: Ax = b X Naive solution x = A−1 b × A−1 may not exist or the system may be inconsistent. =⇒ Generalized Inverse The best approximate solution (least square solution of minimum norm): x = A† b
A† – Moore-Penrose inverse Computational techniques Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
2 / 29
Moore-Penrose Inverse Let A ∈ B(H1 , H2 ). The Moore-Penrose inverse of A is the unique operator A† ∈ B(H2 , H1 ) (if it exists) satisfying 1
AA† A = A
2
A† AA† = A†
3
(AA† )∗ = AA†
4
(A† A)∗ = A† A If A is nonsingular, then A† = A−1 . A† exists if and only if R(A) is closed. Exists for all matrices.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
3 / 29
Remarks Equivalent Definition for A† (i) A† Ax = x, ∀x ∈ R(A∗ ) (ii) A† y = 0, ∀y ∈ N(A∗ ).
Properties of A† R(A† ) = R(A∗ ),
N(A† ) = N(A∗ ),
AA† = PR(A) ,
A† A = PR(A∗ )
PL : – the orthogonal projection onto the space L.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
4 / 29
Motivation There are many physical models where successive computation of matrix inverses must be performed. The amount of computation increases rapidly as the order of the matrix increases. Most cases, the matrix that we deal with may not be the exact matrix representing the data. Truncation and round off errors give rise to perturbations. If we have a nonsingular matrix and its inverse, and suppose that some of the entries of the matrix are altered, then it would be desirable to have an efficient computational method for finding the inverse of the new matrix from the known inverse without computing the inverse of the new matrix, afresh. =⇒ Sherman-Morrison-Woodbury (SMW) formula
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
5 / 29
Sherman-Morrison-Woodbury formula Let A be an n × n and G be an r × r nonsingular matrices with r ≤ n. Also, let V1 and V2 be n × r matrices such that (G −1 + V2 ∗ A−1 V1 )−1 exists. Then (A + V1 GV2∗ )−1 = A−1 − A−1 V1 (G −1 + V2 ∗ A−1 V1 )−1 V2 ∗ A−1 An explicit formula for the inverse of matrices of the form A + V1 G V2 ∗ . Useful: I I I
Situations where r is much smaller than n When A has certain structural properties Computing A−1 V1 (G −1 + V2 ∗ A−1 V1 )−1 V2 ∗ A−1 is ‘easier’ than inverting a general perturbed n × n matrix
Applications: Appears in Statistics, networks, structural analysis, asymptotic analysis, optimization and PDEs1 . 1
W. W. Hager. Updating the inverse of a matrix, SIAM Review, 1989. Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
6 / 29
Objective: Extend SMW formula to operators on Hilbert spaces. Obtain conditions for the nonnegativity of Moore-Penrose inverse of perturbed operators and matrices.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
7 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then • M is a rank-one perturbation of A.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Rank-one Perturbed Operators
Assumptions A ∈ B(H1 , H2 ) with R(A) closed. B = b ⊗ c ∈ B(H1 , H2 ) a rank-one operator (B(x) = hx, ci b, b(6= 0) ∈ H2 and c(6= 0) ∈ H1 ). Define M = A + B. Then • M is a rank-one perturbation of A. • M ∈ B(H1 , H2 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
8 / 29
Mutually Exclusive Cases3,4
For λ = 1 + A† b, c , (i) b ∈ R(A), c ∈ R(A∗ ) and λ = 0. (ii) b ∈ R(A), c ∈ R(A∗ ) and λ 6= 0. (iii) b ∈ / R(A), c ∈ / R(A∗ ). (iv) b ∈ R(A), c ∈ / R(A∗ ). (v) b ∈ / R(A), c ∈ R(A∗ ). Note: The cases are collectively exhaustive. 3
C. D. Meyer, Generalized inversion of modified matrices. SIAM J. Appl. Math., 1973.
4
J. K. Baksalary, O. M. Baksalary, G. Trenkler, A revisitation formulae for the Moore-Penrose inverse of modified matrices. Lin.
Alg. Appl. 2003.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
9 / 29
Notation
d = A† b,
e = (A† )∗ c,
δ = hd, di , η = he, ei , µ = |λ|2 + δψ,
f = PR(A) ⊥ b, φ = hf , f i ,
ν = |λ|2 + ηφ,
g = PR(A∗ ) ⊥ c ψ = hg , g i
¯ + δg ∈ H1 p = λd
q = λe + ηf ∈ H2 where λ = 1 + hd, ci = 1 + hb, ei .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
10 / 29
Case (i) R(M) is closed. Set D : H1 → H1 , E : H2 → H2 by D(x) = hx, di d and E (x) = hx, ei e. Then M † = A† − δ −1 DA† − η −1 A† E + δ −1 η −1 DA† E where d = A† b, e = (A† )∗ c, δ = hd, di , η = he, ei .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
11 / 29
Case (i) R(M) is closed. Set D : H1 → H1 , E : H2 → H2 by D(x) = hx, di d and E (x) = hx, ei e. Then M † = A† − δ −1 DA† − η −1 A† E + δ −1 η −1 DA† E where d = A† b, e = (A† )∗ c, δ = hd, di , η = he, ei .
Case (ii) Assume R(M) is closed. Then M † = A† − λ−1 A† BA† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
11 / 29
Case (iii) Assume that R(M) closed. Set B1 : H2 → H2 by B1 (x) = hx, f i b, G1 : H1 → H1 by G1 (x) = hx, ci g , G2 : H2 → H1 by G2 (x) = hx, f i g . Then M † = A† − φ−1 A† B1 − ψ −1 G1 A† + λφ−1 ψ −1 G2 where f = PR(A) ⊥ b, g = PR(A∗ ) ⊥ c, φ = hf , f i ,
Shani (2013)
Generalized Inverses
ψ = hg , g i .
Prairie Analysis Seminar
12 / 29
Case (iii) Assume that R(M) closed. Set B1 : H2 → H2 by B1 (x) = hx, f i b, G1 : H1 → H1 by G1 (x) = hx, ci g , G2 : H2 → H1 by G2 (x) = hx, f i g . Then M † = A† − φ−1 A† B1 − ψ −1 G1 A† + λφ−1 ψ −1 G2 where f = PR(A) ⊥ b, g = PR(A∗ ) ⊥ c, φ = hf , f i ,
ψ = hg , g i .
Case (iv) Let R(M) be closed. Set G3 : H1 → H1 by G3 (x) = hx, di g . Let D and G1 be defined as in the previous theorems. Then ¯ † BA† ) M † = A† − µ−1 (ψDA† + δG1 A† − λG3 A† + λA where µ = |λ|2 + δψ.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
12 / 29
Case (v) ˜ Set B(x) = hx, f i b and E˜ (x) = hx, f i e. Let E be as before. Let R(M) be closed. Then ˜ − λA† E˜ + λA ¯ † BA† ) M † = A† − ν −1 (φA† E + ηA† B where ν = |λ|2 + ηφ.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
13 / 29
Case (v) ˜ Set B(x) = hx, f i b and E˜ (x) = hx, f i e. Let E be as before. Let R(M) be closed. Then ˜ − λA† E˜ + λA ¯ † BA† ) M † = A† − ν −1 (φA† E + ηA† B where ν = |λ|2 + ηφ. The above 5 cases are generalized to obtain formulae for (A + V1 GV2∗ )† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
13 / 29
Example Let H1 = H2 = B(l 2 ) and A ∈ B(l 2 ) be the right shift operator. Then • A† = A∗ - the left shift operator. • B(x) = hx, ci b, for fixed nonzero vectors b, c ∈ l2 and M = A + B. • R(M) is closed5 if kbk kck < 1. Take b = e1 and c = e3 where ei ∈ l 2 is the vector containing i th element 1 and all other elements zero. b∈ / R(A) and c ∈ R(A∗ ) M † (x) = A† x − 21 (x4 − x1 )e3 , where x = (x1 , x2 , x3 , . . .) ∈ l 2 . 5
O. Christensen, Operators with closed range, pseudo-Inverses, and perturbation of frames for a subspace. Canad. Math.
Bull.,1999.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
14 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2 Then ˆ is closed. X R(Ω)
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Formulae for (A + V1 GV2∗ )† Generalization of Case (i) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H2 ), V2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ), V2∗ A† V1 = −IH2 . ˆ = A + V1 V ∗ . • Ω 2 Then ˆ is closed. X R(Ω) ˆ † = PR(A† V )⊥ A† P † ∗ ⊥ X Ω R(A V2 ) 1
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
15 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ )
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ ) • Ω = A + V1 GV2∗ have closed range space.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of case (ii) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ), G ∈ B(H2 , H1 ). • R(G ) and R(G † + V2∗ A† V1 ) are closed. • R(V1 ) ⊆ R(A), R(V2 ) ⊆ R(A∗ ). • R(V1∗ ) ⊆ R((G † + V2∗ A† V1 )∗ ), R(V2∗ ) ⊆ R(G † + V2∗ A† V1 ), R(V1∗ ) ⊆ R(G ), R(V2∗ ) ⊆ R(G ∗ ) • Ω = A + V1 GV2∗ have closed range space. Then, Ω† = A† − A† V1 (G † + V2∗ A† V1 )† V2∗ A† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
16 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ). • Ω = A + (U1 + W1 )G (U2 + W2 )∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iii) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), R(U1 ) ⊆ R(A), R(W1 ) ⊆ R(A)⊥ . • U2 , W2 ∈ B(H2 , H1 ), R(U2 ) ∈ R(A∗ ), R(W2 ) ∈ R(A∗ )⊥ . • G ∈ B(H2 , H1 ), R(G ) is closed. • R((U1 + W1 )∗ ) ⊆ R(G ), R((U2 + W2 )∗ ) ⊆ R(G ∗ ), R(G ∗ ) ⊆ R(W2∗ ), R(G ) ⊆ R(W1∗ ). • Ω = A + (U1 + W1 )G (U2 + W2 )∗ with R(Ω) closed. Then ∗
∗
Ω† = A† − W2 † U2 ∗ A† − A† U1 W1 † + W2 † (G † + U2∗ A† U1 )W1 † .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
17 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L. • Ω = A + V1 (U2 + W2 )∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (iv) • A ∈ B(H1 , H2 ) with closed range space. • V1 ∈ B(H1 , H2 ), U2 , W2 ∈ B(H2 , H1 ). • R(V1 ) ⊆ R(A), R(U2 ) ⊆ R(A∗ ), R(W2 ) ⊆ R(A∗ )⊥ . • W2 ∗ W2 = I , (A† V1 )∗ (A† V1 ) = I , L = I + U2∗ A† V1 . • L∗ L = LL∗ , K = I + L∗ L. • Ω = A + V1 (U2 + W2 )∗ with R(Ω) closed. Then Ω† = A† − (A† V1 )K −1 (A† V1 )∗ A† − W2 K −1 U2 ∗ A† + W2 K −1 L(A† V1 )∗ A† − (A† V1 )K −1 L∗ U2 ∗ A† .
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
18 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 )
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
• Ω = A + (U1 + W1 )V2 ∗ with R(Ω) closed.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Generalization of Case (v) • A ∈ B(H1 , H2 ) with R(A) closed. • U1 , W1 ∈ B(H1 , H2 ), V2 ∈ B(H2 , H1 ) • R(U1 ) ⊆ R(A), R(W2 ) ⊆ R(A)⊥ , R(V2 ) ⊆ R(A∗ ). ∗ ∗ ˜ = I + V2 ∗ A† U1 . • W1 ∗ W1 = I , (A† V2 )∗ (A† V2 ) = I , L ˜∗ L ˜=L ˜L ˜∗ , N = I + L ˜∗ L. ˜ • L
• Ω = A + (U1 + W1 )V2 ∗ with R(Ω) closed. Then ∗
∗
Ω† = A† − A† (A† V2 )N −1 (A† V2 )∗ − A† U1 N −1 W1 ∗ ∗ ˜ −1 W1 ∗ − A† U1 L ˜∗ N −1 (A† ∗ V2 )∗ . + A† A† V2 LN
Certain applications of these formulae are obtained. Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
19 / 29
Example 1
• H = l 2 , A be the right shift operator on H. • X , Y ∈ B(H) be s. t. −X (x1 , x2 , x3 , . . .) = (0, 0, 0, x1 , x2 , x3 , . . .) Y (x1 , x2 , x3 , . . .) = (x1 , x2 , x1 , x2 , x3 , x4 , . . .). • Set Ω = A + XY ∗ . X R(X ) ⊆ R(A), R(Y ) ⊆ R(A∗ ) and −Y ∗ A† X = I – case (i). X Ω† (x1 , x2 , . . .) = (
Shani (2013)
x2 − x4 x3 − x5 , , 0, 0, . . .). 2 2
Generalized Inverses
Prairie Analysis Seminar
20 / 29
Example 2 • H = l 2 and A ∈ B(H) be A(x1 , x2 , . . .) = (x1 , 0, x2 , 0, x3 , 0, . . .). • R(A) is a closed subspace of l 2 . and A† (x) = A∗ (x) = (x1 , x3 , x5 , . . .) for all x = (x1 , x2 , x3 , . . .). • Let X (x1 , x2 , . . .) = (x1 , 0, x2 , 0, 0, 0, . . .) and Y (x1 , x2 , . . .) = (x1 , x3 , 0, 0, 0, . . .) for all x ∈ H • Ω = A + XY ∗ . X R(Ω) is closed. X Both X , Y ∈ B(H) and R(X ) ⊆ R(A), R(Y ) ⊆ R(A∗ ) and (I + Y ∗ A† X )(x1 , x2 , x3 , . . .) = (2x1 , x2 , x3 + x2 , x4 , x5 , . . .) is bijective. X Ω† (x) = (A† − A† X (I + Y ∗ A† X )−1 Y ∗ A† )(x) = (
Shani (2013)
Generalized Inverses
x1 , x3 , x5 , x7 , x9 , . . .). 2
Prairie Analysis Seminar
21 / 29
Application • H1 = H2 = l2 and A be the right shift operator defined on l 2 . • Then R(A) is closed and hence it has a bounded Moore-Penrose inverse. • A† x = A∗ x = (x2 , x3 , x4 ...), for every x = (x1 , x2 , x3 , ...) ∈ l2 , the left shift operator on l2 . • A ≥ 0 and A† ≥ 0. • Let b = he2 and c = −e1 with h ∈ / {0, 1}. • Take M = A + B, where B = h hx, −e1 i e2 . • R(M) is closed.
• b ∈ R(A), c ∈ R(A∗ ) and λ = 1 + A† b, c = 1 − h 6= 0. Thus we are in Case (ii). • M † (x) = A† (x) − X M † ≥ 0 iff h ∈ Shani (2013)
h 1−h x2 e1 . (−∞, 21 )\{0}
∪ (1, ∞)
Generalized Inverses
Prairie Analysis Seminar
22 / 29
Star partial order and rank-one perturbation Let A1 , A2 ∈ B(H) with R(A1 ) and R(A2 ) closed. The star partial order relating A1 and A2 is defined as ∗
A1 ≤ A2 if and only if A1 A1 † = A2 A1 † and A1 † A1 = A1 † A2 .
To verify whether a rank-one perturbation preserves the star partial order ∗
Let A1 ≤ A2 . b and c in H and B(x) = hx, ci b. Set M1 = A1 + B and M2 = A2 + B. ∗
Let A1 , A2 ∈ B(H1 , H2 ) such that A1 ≤ A2 . Let B = b ⊗ c where b ∈ H2 , ∗
c ∈ H1 and M1 = A1 + B, M2 = A2 + B. Then M1 ≤ M2 if b ∈ R(A1 ) and c ∈ R(A∗1 ). Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
23 / 29
Remark Rank-one perturbations do not preserve star partial order for cases (iii), (iv) and (v) even in the finite dimensional case. Examples: A1 =
1 0 0 0
and A2 =
1 0 0 1
.
∗
Then we have A1 ≤ A2 . 0 • Take b = c = . Then b ∈ / R(A1 ) and c ∈ / R(A∗1 ) and this 1 1 0 1 0 corresponds to case (iii). Here M1 = and M2 = . 0 1 0 2 ∗
But M1 M1T 6= M2 M1T and so M1 ≤ M2 does not hold.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
24 / 29
−1 0 and c = . Here, b ∈ R(A1 ) and c ∈ / R(A∗1 ), 0 1 and so we are in case (iv). Now, consider the rank-one perturbations M1 = A1 + bc T and M2 = A + bc T . Then, 1 −1 1 −1 M1 = and M2 = . 0 0 0 1
• Take b =
Here again, we have M1 M1T 6= M2 M1T . • By interchanging the roles of b, c and A with c, b and A∗ , respectively, it follows that the result does not hold in case (v).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
25 / 29
Theorem ∗
A1 ≤ A2 . B(x) = hx, ci b for fixed nonzero vectors b, c ∈ H. M1 = A1 + B and M2 = A2 + B. Then any two of the following imply the third: ∗
(a) A1 ≤ A2 ∗
(b) M1 ≤ M2 (c) R(B) ⊆ N(A2 − A1 )∗ and R(B ∗ ) ⊆ N(A2 − A1 ).
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
26 / 29
Shani Jose and K. C. Sivakumar, Moore-Penrose inverses of perturbed operators on Hilbert spaces, Combinatorial Matrix Theory and Generalized Inverses of Matrices, Springer, India (2013) 119–131. Shani Jose and K. C. Sivakumar, On Nonnegative Moore-Penrose Inverses of Perturbed Matrices. J. Appl. Math., (2013), Article ID 680975, 7 pages.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
27 / 29
References J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, The Ann. Math. Statist., 21 (1950), 124–127. W. W. Hager, Updating the inverse of a matrix, SIAM Review, 31 (1989), 221–239. K. S. Riedel, A Sherman-Morrison-Woodbury identity for rank augmenting matrices with application to centering, SIAM J. Matrix Anal. Appl., 13 (1992), 659–662. J. K. Baksalary, O. M. Baksalary, and G. Trenkler, A revisitation formulae for the moore-penrose inverse of modified matrices. Linear Algebra and Applications, 372 (2003), 207–224.
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
28 / 29
Thank you
Shani (2013)
Generalized Inverses
Prairie Analysis Seminar
29 / 29
