How does one compute the left eigenvectors of a matrix? I cannot seem to quite get the answer.. I don't care what the matrix is. Let's just say I have matrix $A$ and have found the 'right' eigenvectors $e$ and I want to compute the left eigenvectors. Do we $A^Te = c$ Then the left eigenvector is $c^T$? I did this for a 2x2 and it seemed to work, but for a 3x3 it did not (according to the answers in the book).
Thanks in advance!
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$\begingroup$In general, the left (resp. right) eigenvectors of a matrix $A$ are the right (resp. left) eigenvectors of the matrix $A^T$. It follows quite straightforwardly that the eigenvalues of $A$ and those of $A^T$ coincide (including multiplicities) but the same is not true of the left and right eigenvectors. Generally, one simply needs to compute the, say, right eigenvectors even if one already has the left eigenvectors. Of course, for special matrices (like symmetric ones), left and right eigenvectors do coincide.
$\endgroup$ 2 $\begingroup$With $ uA = ku $ you do this
$ (u A ) ^* = k^* u^* $ where * means complex conjugate, and you get $ A^* u^* = k^* u^* $
That should do it
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