Deflation: Suppose A has eigenvalue λ and corresponding eigenvector v.
(a) Let b be any vector. Prove that the matrix B = A - vbT also has v as an eigenvector, now with eigenvalue λ - β where β = v €¢ b.
(b) Prove that if μ ‰  λ - β is any other eigenvalue of A, then it is also an eigenvalue of B.
(c) Given a nonsingular matrix A with eigenvalues λ1, λ2, ... , λn and λ1 ‰  λj, j ‰¥ 2, explain how to construct a deflated matrix B whose eigenvalues are 0, λ2, ..., λn.
(d) Try out your method on the matrices

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