(Rank−1 Updates of Linear Systems)
(a) Set
Use the matrix M to solve the system
(b) Consider now a new system
The matrices C and A differ by the rank−1 matrix E. Use MATLAB to verify that the rank of E is 1. Use MATLABS’s “\” operation to solve the system
(c) Let us now solve
and then compute the solution x by
Compute the residual vector
(d) To see why the rank−1 update method works, use MATLAB to compute and compare
Cy and
Prove that if all computations had been carried out in exact arithmetic, these two
Cz and
Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Use these identities to prove that
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