Steps
We assume that the matrices are symmetric and positive definite and assume we have scaled the eigenvectors such that
where is the Kronecker delta.
Now we want to solve the equation
Substituting, we get
which expands to
Canceling from (1) leaves
Removing the higher-order terms, this simplifies to
When the matrix is symmetric, the unperturbed eigenvectors are orthogonal and so we use them as a basis for the perturbed eigenvectors. That is, we want to construct
where the are small constants that are to be determined. Substituting (4) into (3) and rearranging gives
Or:
By equation (1):
Because the eigenvectors are orthogonal, we can remove the summations by left multiplying by :
By use of equation (1) again:
The two terms containing are equal because left-multiplying (1) by gives
Canceling those terms in (6) leaves
Rearranging gives
But by (2), this denominator is equal to 1. Thus
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Then, by left-multiplying equation (5) by (for ):
Or by changing the name of the indices:
To find, use
Read more about this topic: Eigenvalue Perturbation
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