In mathematics, eigenvalue perturbation is a perturbation

Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

 approach to finding eigenvalues and eigenvectors of systems perturbed from one with known eigenvectors and eigenvalues. It also allows one to determine the sensitivity of the eigenvalues and eigenvectors with respect to changes in the system. The following derivations 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,


That is, we know and for . Now suppose we want to change the matrices by a small amount. That is, we want to let


and


where all of the terms are much smaller than the corresponding term. We expect answers to be of the form


and

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
   ■

Then, by left multiplying equation (6) by (for ):


Or by changing the name of the indices:


To find , use

Summary

and

for infinetesimal and (the high order terms in (3) being negligible)

Results

This means it is possible to efficiently do a sensitivity analysis

Sensitivity analysis

Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...

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.)


and


Similarly


and