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. This article is focused on the case of the perturbation of a simple eigenvalue, as opposed to a multiplicity of eigenvalues.

Motivation for generalized eigenvalues Many scientific fields use eigenvalues to obtain solutions. Generalized eigenvalue problems are less widespread but are key in the study of vibrations. They are useful when the Galerkin or Rayleigh-Ritz methods are used to find approximate solutions of partial differential equations modeling vibrations of structures such as strings and plates - Courant (1943) is fundamental. The finite element method is a widespread particular case.

In classical mechanics, generalized eigenvalues may crop up when inspecting vibrations of multiple degrees of freedom systems close to equilibrium. In this case the kinetic energy provides the mass matrix , the potential strain energy provides the rigidity matrix .

With both methods, the following system of differential equations or matrix differential equation is derived: with the mass matrix , the damping matrix and the rigidity matrix . If the damping effect is neglected, , and a solution of the form is assumed, and are obtained as solutions to the generalized eigenvalue problem .

Setting of perturbation for a generalized eigenvalue problem Suppose the solutions to the generalized eigenvalue problem are known to be

where and are matrices. That is, we know the eigenvalues and eigenvectors for . An important note is that the eigenvalues are required to be distinct.

In order to perturb the matrices, one must find the eigenvalues and eigenvectors of

where

with the perturbations and much smaller than and respectively. Then the new eigenvalues and eigenvectors are expected to be similar to the original, plus small perturbations:

Steps Under the assumption that the matrices are symmetric, positive definite, and assume the eigenvectors are scaled such that

where is the Kronecker delta. Now the equation to be solved is

In this article, the study is restricted to first order perturbation.

First order expansion of the equation Substituting in (1) results in

which expands to

Canceling from (0) () leaves

Removing the higher-order terms, this simplifies to

In other words, no longer denotes the exact variation of the eigenvalue but its first order approximation.

As the matrix is symmetric, the unperturbed eigenvectors are orthogonal and so can be used as a basis for the perturbed eigenvectors. This is the same as

with , where are small constants that are to be determined.

In the same way, substituting in (2), and removing higher order terms,

The derivation is then split into two paths.

First path: get first eigenvalue perturbation Eigenvalue perturbation Starting with (3)which is then left multiplied with along with using (2) as well as its first order variation (5); one gets

This is the first order perturbation of the generalized Rayleigh quotient with fixed :

Moreover, for , the formula should be compared with Bauer-Fike theorem which provides a bound for eigenvalue perturbation.

Eigenvector perturbation One then left multiplies (3) with for and get

Recalling that for , one may substitute for

As the eigenvalues are assumed to be simple, for

Moreover (5) (the first order variation of (2) ) yields

All of the components of have now been obtained.

Second path: Straightforward manipulations Substituting (4) into (3) and rearranging gives

Because the eigenvectors are -orthogonal when is positive definite, one 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, as for (this is the assumption of simple eigenvalues) by left-multiplying equation (5) by :

Or by changing the name of the indices:

To find , using the fact that

implies:

Summary of the first order perturbation result In the case where all the matrices are Hermitian positive definite and all the eigenvalues are distinct,

for infinitesimal and (the higher order terms in (3) being neglected).

A proof that higher order terms may be neglect may be derived using the implicit function theorem.

Theoretical derivation

Perturbation of an implicit function. In the next paragraph, we shall use the Implicit function theorem (Statement of the theorem ); we notice that for a continuously differentiable function , with an invertible Jacobian matrix , from a point solution of , we get solutions of with close to in the form where is a continuously differentiable function; moreover the Jacobian marix of is provided by the linear system . As soon as the hypothesis of the theorem is satisfied, the Jacobian matrix of may be computed with a first order expansion of , we get

; as , it is equivalent to equation .

Eigenvalue perturbation: a theoretical basis. We use the previous paragraph (Perturbation of an implicit function) with somewhat different notations suited to eigenvalue perturbation; we introduce , with with . In order to use the Implicit function theorem, we study the invertibility of the Jacobian with

. Indeed, the solution of

may be derived with computations similar to the derivation of the expansion.

When is a simple eigenvalue, as the eigenvectors form an orthonormal basis, for any right-hand side, we have obtained one solution therefore, the Jacobian is invertible.

The implicit function theorem provides a continuously differentiable function

hence the expansion with little o notation:

. with

This is the first order expansion of the perturbed eigenvalues and eigenvectors. which is proved.

Results of sensitivity analysis with respect to the entries of the matrices The 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

Why generalized eigenvalues? In the entry applications of eigenvalues and eigenvectors we find numerous scientific fields in which eigenvalues are used to obtain solutions. Generalized eigenvalue problems are less widespread but are a key in the study of vibrations. They are useful when we use the Galerkin method or Rayleigh-Ritz method to find approximate solutions of partial differential equations modeling vibrations of structures such as strings and plates; the paper of Courant (1943) is fundamental. The Finite element method is a widespread particular case.

In classical mechanics, generalized eigenvalues may crop up when we look for vibrations of multiple degrees of freedom systems close to equilibrium; the kinetic energy provides the mass matrix , the potential strain energy provides the rigidity matrix . For further details, see the first section of this article of Weinstein (1941, in French)

With both methods, we obtain a system of differential equations or Matrix differential equation with the mass matrix , the damping matrix and the rigidity matrix . If we neglect the damping effect, we use , we can look for a solution of the following form ; we obtain that and are solution of the generalized eigenvalue problem

Setting of perturbation for a generalized eigenvalue problem Suppose we have solutions to the generalized eigenvalue problem,

where and 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 and much smaller than and 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

In this article we restrict the study to first order perturbation.

First order expansion of the equation Substituting in (1), we get

which expands to

Canceling from (0) () leaves

Removing the higher-order terms, this simplifies to

In other words, no longer denotes the exact variation of the eigenvalue but its first order approximation.

As 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

with , where the are small constants that are to be determined.

In the same way, substituting in (2), and removing higher order terms, we get

The derivation can go on with two forks.

First fork: get first eigenvalue perturbation Eigenvalue perturbation We start with (3) we left multiply with and use (2) as well as its first order variation (5); we get

We notice that it is the first order perturbation of the generalized Rayleigh quotient with fixed :

Moreover, for , the formula should be compared with Bauer-Fike theorem which provides a bound for eigenvalue perturbation.

Eigenvector perturbation We left multiply (3) with for and get

We use for .

As the eigenvalues are assumed to be simple, for

Moreover (5) (the first order variation of (2) ) yields

We have obtained all the components of .

Second fork: Straightforward manipulations 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 :