Eigenvalue perturbation

In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system. 
This type of analysis was popularized by Lord Rayleigh, in his investigation of harmonic vibrations of a string perturbed by small inhomogeneities. 
The derivations in this article are essentially self-contained and can be found in many texts on numerical linear algebra or numerical functional analysis.
Example.
Suppose we have solutions to the generalized eigenvalue problem,
where formula_2 and formula_3 are matrices. That is, we know the eigenvalues and eigenvectors for . It is also required that the eigenvalues are distinct. Now suppose we want to change the matrices by a small amount. That is, we want to find the eigenvalues and eigenvectors of
where
with the perturbations formula_6 and formula_7 much smaller than formula_8 and formula_9 respectively. Then we expect the new eigenvalues and eigenvectors to be similar to the original, plus small perturbations:
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 (0) (formula_15) 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
Because the eigenvectors are -orthogonal when is positive definite, we can remove the summations by left-multiplying by formula_20:
By use of equation (1) again:
The two terms containing are equal because left-multiplying (1) by formula_20 gives
Canceling those terms in (6) leaves
Rearranging gives
But by (2), this denominator is equal to 1. Thus
Then, by left-multiplying equation (5) by formula_28:
Or by changing the name of the indices:
To find , use the fact that:
implies:
Summary.
for infinitesimal formula_6 and formula_7 (the high order terms in (3) being negligible)
Results.
This means it is possible to efficiently do a sensitivity analysis on as a function of changes in the entries of the matrices. (Recall that the matrices are symmetric and so changing will also change , hence the term.)
Similarly
Existence of eigenvectors.
Note that in the above example we assumed that both the unperturbed and the perturbed systems involved symmetric matrices, which guaranteed the existence of formula_38 linearly independent eigenvectors. An eigenvalue problem involving non-symmetric matrices is not guaranteed to have formula_38 linearly independent eigenvectors, though a sufficient condition is that formula_8 and formula_9 be simultaneously diagonalizable. 