Are the eigenvectors of A and A^T related?

[Leo321]

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 A^T associated with $$\lambda$$ or is it completely independent of v?

 

[AlephZero]

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)