## Eigenvalue SensitivityFor an matrix with elements , eigenvalues , (right) eigenvectors satisfying , and left eigenvectors satisfying , it is easy to obtain an expression for the sensitivity of to small changes in the element . (See [Horn and Johnson, 1990] and http://www.astro.virginia.edu/~eww6n/math/): so where and are the -th and -th components of and . The random complex matrix
has eigenvalues
and (right) eigenvectors
The left eigenvectors are computed using since if .
Introducing the diagonal matrix  of eigenvalues,
we see that ,
and :
Moreover, the dot products of all possible pairs of left and right eigenvectors forms a matrix , which is diagonal (since ).
The diagonal entries are the products .
If we perturb by 0.01+0.02i,
the matrix becomes
and the perturbed eigenvalues are
Defining
we can estimate the perturbed values from .
The results are in good agreement.
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