Are the eigenvectors of A and A^T related?

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I have an (unknown) matrix A and with real non-negative values. I know its largest eigenvalue \lambda and the associated eigenvector, v. (I know nothing about the other eigenvectors). Does this give me any information about the eigenvector of AT associated with \lambda or is it completely independent of v?
 
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The eigenvalues of A and A^T are the same. Consider the characteristic polynomial as the expansion of the determinant of A. The characteristic polynomials of A and A^T are identical so they have the same set of roots.

However you can't say anything about the left and right eigenvectors. For example let A =

2 1
3 0

Eigenvalues are 3 and -1
Correponding vectors are (1 1) and (1 -3)

A^T =
2 3
1 0

Eigenvalues are again 3 and -1
Corresponding eigenvectors are (3 1) and (1 -1)
 

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