Let v = x + i y be an eigenvector corresponding to a complex, non-real eigenvalue of
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Let v = x + i y be an eigenvector corresponding to a complex, non-real eigenvalue of the real n × n matrix A.
(a) Prove that the Krylov subspaces V(k) for k ≥ 2 generated by both x and y are all two-dimensional.
(b) Is the converse valid? Specifically, if dim V(3) = 2, then all V(k) are two dimensional for k ≥ 1 and spanned by the real and imaginary parts of a complex eigenvector of A.
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a Since v is an eigenvector of A we have Av v where is the complex nonreal eigenvalue Then for k 2 w...View the full answer
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