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Asked Feb 11, 2020
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Let Z be a cycle of generalized eigenvectors of a linear operator T on V that
corresponds to the eigenvalue 2 Prove that span(Z) is a T-invariant
subspace of V.
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Let Z be a cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eigenvalue 2 Prove that span(Z) is a T-invariant subspace of V.

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Hi! This is an Advanced Math question. We are answering it now but kindly post such questions under Advanced Math now onward.

Here, we are given that Z is the cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eigenvalue λ.

Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T. Suppose that

Z = { v1, v2, ... ,vn }

Or

if p is is the smallest positive integer such that (T- λI)pv = 0, Z = {(T- λI)p-1v, (T- λI)p-2v, ... , (T- λI)v, v},  

cycles of generalized eigenvectors of T corresponding to λ.

Note:

The generalized eigenvectors in a chain are linearly independent.

 

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