(a) Consider matrices A, U, and V , where A is an invertible n × n matrix, and U and V are n × k matrices with rank k < n. Prove the Sherman–Morrison–Woodbury formula, i.e., that T = I − (V^T) (A^−1)U is nonsingular if and only ifA – UV^T is nonsingular, in which case (A – UV^T)^−1 = (A^−1)−( A^−1)(UT^−1)(V^T)A^−1. (b) Suppose you have a fast algorithm for solving Ax = b (for example, using an LU factorisation of A). Show how to build a fast algorithm for solving Bx = c, where B = A – UV^T.
(a) Consider matrices A, U, and V , where A is an invertible n × n matrix, and U and V are n × k matrices with rank k < n. Prove the Sherman–Morrison–Woodbury formula, i.e., that T = I − (V^T) (A^−1)U is nonsingular if and only ifA – UV^T is nonsingular, in which case (A – UV^T)^−1 = (A^−1)−( A^−1)(UT^−1)(V^T)A^−1. (b) Suppose you have a fast algorithm for solving Ax = b (for example, using an LU factorisation of A). Show how to build a fast algorithm for solving Bx = c, where B = A – UV^T.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
(a) Consider matrices A, U, and V , where A is an invertible n × n matrix, and U and V are n × k matrices with rank k < n. Prove the Sherman–Morrison–Woodbury formula, i.e., that T = I − (V^T) (A^−1)U is nonsingular if and only ifA – UV^T is nonsingular, in which case (A – UV^T)^−1 = (A^−1)−( A^−1)(UT^−1)(V^T)A^−1.
(b) Suppose you have a fast algorithm for solving Ax = b (for example, using an LU factorisation of A). Show how to build a fast algorithm for solving Bx = c, where B = A – UV^T.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
