# principal eigen vector of a matrix

Given a matrix $M$ and given its principal eigen vector $E$, is there any way we can find the eigen vector $E_i$ of the matrix $M_i$. Where $M_i$ is defined as matrix $M$ with $i$th row and $i$th column deleted.

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What if $E = e_1$? Then you have practically no information whatsoever about $M_1$ (applying orthogonal transformation to $M_1$ and sticking it back into the original matrix does not change $E$). –  Willie Wong Sep 29 '10 at 19:40
You may wish to seek some references on "Matrix Completion" techniques. Depending on the rank of M, I suppose the answer is "it depends." –  Ed Gorcenski Sep 29 '10 at 20:31