Is there any ways to compute the eigen vector without computing explicitly the associated eigenvalue?
Actually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, so I have to know its eigenvector first.
Is there any ways to compute the eigen vector without computing explicitly the associated eigenvalue? Actually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, so I have to know its eigenvector first. 

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If you pick a random vector $v$ and look at $v_n=A^n v/\ A^n v\,$ that will converge to the dominant eigenvector. 


What do you know about the matrix? If we know that the rows all have the same sum (but not what that sum is) then we would essentially find it by multiplying by the corresponding eigenvector, $\mathbb{j}$, the all $1$'s vector. This will be the largest eigenvalue provided that the entries are nonnegative. One way this could happen (but not the only one) is if the rows are identical or merely each is a permutation of the first. If certain rows are equal then we know that $0$ is a eigenvalue although we never "computed" it. Then we do know an eigenvector. In a circulant matrix we know all the eigenvectors (not just $\mathbb{j}$) and we essentially use them to compute the corresponding eigenvalues. 

