Eigenvalue Sensitivity

For 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 .

has eigenvalues

and (right) eigenvectors

The left eigenvectors are computed using since if  .

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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