Show that A and A^{T} have the same eigenvalues. what if anything can we say about the associated eigenvectors of A and A^{T}?
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The eigenvectors of are generally different to those of . However, there is a relationship between the two:
If is an eigenvector of corresponding to the eigenvalue and is an eigenvector of corresponding to the eigenvalue then .
Proof:
.... (1)
.... (2)
(1) - (2): and the result is easily seen.