Fanaticism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Fanaticism

The Fanatics of Tangier by Eugène Delacroix, Minneapolis Institute of Arts

Fanaticism is a belief or behavior involving uncritical zeal or an obsessive enthusiasm. The political theorist Zachary R. Goldsmith provides a "cluster account" of the concept of fanaticism, identifying ten main attributes that, in various combinations, constitute it: messianism, inappropriate relationship to reason (irrationality), an embrace of abstraction, a desire for novelty, the pursuit of perfection, an opposition to limits, the embrace of violence, absolute certitude, excessive passion, and an attractiveness to intellectuals.

Definitions

Etienne-Pierre-Adrien Gois, Voltaire defending Innocence against Fanaticism, c. 1791

Philosopher George Santayana defines fanaticism as "redoubling your effort when you have forgotten your aim". The fanatic displays very strict standards and little tolerance for contrary ideas or opinions. Tõnu Lehtsaar has defined the term fanaticism as the pursuit or defence of something in an extreme and passionate way that goes beyond normality. Religious fanaticism is defined by blind faith, the persecution of dissidents and the absence of reality.

Causes

Japanese holdouts persisted on various islands in the Pacific Theatre until at least 1974. Hiroo Onoda offering his military sword on the day of his surrender.

Fanaticism is a result from multiple cultures interacting with one another. Fanaticism occurs most frequently when a leader makes minor variations on already existing beliefs, which then drives the followers into a frenzy. In this case, fanaticism is used as an adjective describing the nature of certain behaviors that people recognize as cult-like. Margaret Mead referred to the style of defense used when the followers are approached. The most consistent thing presented is the priming, or preexisting, conditions and mind state needed to induce fanatical behavior. Each behavior is obvious once it is pointed out; a closed mind, no interest in debating the subject of worship, and over reaction to people who do not believe.

In his book Crazy Talk, Stupid Talk, Neil Postman states that "the key to all fanatical beliefs is that they are self-confirming....(some beliefs are) fanatical not because they are 'false', but because they are expressed in such a way that they can never be shown to be false."

Similar behaviors

The behavior of a fan with overwhelming enthusiasm for a given subject is differentiated from the behavior of a fanatic by the fanatic's violation of prevailing social norms. Though the fan's behavior may be judged as odd or eccentric, it does not violate such norms. A fanatic differs from a crank, in that a crank is defined as a person who holds a position or opinion which is so far from the norm as to appear ludicrous and/or probably wrong, such as a belief in a Flat Earth. In contrast, the subject of the fanatic's obsession may be "normal", such as an interest in religion or politics, except that the scale of the person's involvement, devotion, or obsession with the activity or cause is abnormal or disproportionate to the average.

Types

• Consumer fanaticism – the level of involvement or interest one has in the liking of a particular person, group, trend, artwork or idea
• Emotional fanaticism
• Ethnic or racial supremacist fanaticism
• Leisure fanaticism – high levels of intensity, enthusiasm, commitment and zeal shown for a particular leisure activity
• Nationalistic or patriotic fanaticism
• Political, ideological fanaticism.
• Religious fanaticism – considered by some to be the most extreme form of religious fundamentalism. Entail promoting religious point of views
• Sports fanaticism – high levels of intensity surrounding sporting events. This is either done based on the belief that extreme fanaticism can alter games for one's favorite team (Ex: Knight Krew), or because the person uses sports activities as an ultra-masculine "proving ground" for brawls, as in the case of football hooliganism.

Evolutionary ethics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Evolutionary_ethics 

Evolutionary ethics is a field of inquiry that explores how evolutionary theory might bear on our understanding of ethics or morality. The range of issues investigated by evolutionary ethics is quite broad. Supporters of evolutionary ethics have argued that it has important implications in the fields of descriptive ethics, normative ethics, and metaethics.

Descriptive evolutionary ethics consists of biological approaches to morality based on the alleged role of evolution in shaping human psychology and behavior. Such approaches may be based in scientific fields such as evolutionary psychology, sociobiology, or ethology, and seek to explain certain human moral behaviors, capacities, and tendencies in evolutionary terms. For example, the nearly universal belief that incest is morally wrong might be explained as an evolutionary adaptation that furthered human survival.

Normative (or prescriptive) evolutionary ethics, by contrast, seeks not to explain moral behavior, but to justify or debunk certain normative ethical theories or claims. For instance, some proponents of normative evolutionary ethics have argued that evolutionary theory undermines certain widely held views of humans' moral superiority over other animals.

Evolutionary metaethics asks how evolutionary theory bears on theories of ethical discourse, the question of whether objective moral values exist, and the possibility of objective moral knowledge. For example, some evolutionary ethicists have appealed to evolutionary theory to defend various forms of moral anti-realism (the claim, roughly, that objective moral facts do not exist) and moral skepticism.

History

The first notable attempt to explore links between evolution and ethics was made by Charles Darwin in The Descent of Man (1871). In Chapters IV and V of that work Darwin set out to explain the origin of human morality in order to show that there was no absolute gap between man and animals. Darwin sought to show how a refined moral sense, or conscience, could have developed through a natural evolutionary process that began with social instincts rooted in our nature as social animals.

Not long after the publication of Darwin's The Descent of Man, evolutionary ethics took a very different—and far more dubious—turn in the form of Social Darwinism. Leading Social Darwinists such as Herbert Spencer and William Graham Sumner sought to apply the lessons of biological evolution to social and political life. Just as in nature, they claimed, progress occurs through a ruthless process of competitive struggle and "survival of the fittest," so human progress will occur only if government allows unrestricted business competition and makes no effort to protect the "weak" or "unfit" by means of social welfare laws. Critics such as Thomas Henry Huxley, G. E. Moore, William James, Charles Sanders Peirce, and John Dewey roundly criticized such attempts to draw ethical and political lessons from Darwinism, and by the early decades of the twentieth century Social Darwinism was widely viewed as discredited.

The modern revival of evolutionary ethics owes much to E. O. Wilson's 1975 book, Sociobiology: The New Synthesis. In that work, Wilson argues that there is a genetic basis for a wide variety of human and nonhuman social behaviors.

More recently, a number of evolutionary biologists, including Richard Alexander, Robert Trivers, and George Williams, have argued for a different relation between ethics and evolution. In Alexander's words: “Ethical questions, and the study of morality or concepts of justice and right and wrong, derive solely from the existence of conflicts of interest.”  The latter, in turn, are inevitable consequences of genetic individuality. Alexander argued that "Because morality involves conflicts of interest, it cannot easily be generalized into a universal despite virtually continual efforts by utilitarian philosophers to do that; morality does not derive its meaning from sets of universals or undeniable facts." Rather, he argued,

The two major contributions that evolutionary biology may be able to make to this problem are, first, to justify and promote the conscious realization that it is conflicts of interest concentrated at the individual level which lead to ethical questions, and, second, to help identify the nature and intensity of the conflicts of interest involved in specific cases.

This view runs contrary to that of the majority of philosophers who work on evolutionary ethics, since it denies the existence of an innate “moral sense” in humans.

As an example of genetic conflict, parents are selected to direct their time and resources equally among their offspring, but any particular child is more strongly related to itself than to any of its siblings, and so will desire a greater amount of parental investment than either parent is selected to give. A consequence of this parent-offspring conflict is that natural selection is unable to instill a universal sense of what is "just" or "fair" with regard to treatment of siblings, since behavior that is most conducive to propagation of the parents' genes differs from what is most favorable for the child's genes.

Alexander noted that a focus on conflicts of interest is common among biologists and other non-philosophers, but that "many moral philosophers do not approach the problem of morality and ethics as if it arose as an effort to resolve conflicts of interests." He defined what he called "moral systems" as societal (not evolved) responses to conflicts of interest. Among other examples, he cited societal rules or laws imposing monogamy. The behavioral conflicts that are addressed by such rules have their evolutionary origin in the (genetic) sexual conflict between men and women.

Descriptive evolutionary ethics

See also: Evolution of morality

The most widely accepted form of evolutionary ethics is descriptive evolutionary ethics. Descriptive evolutionary ethics seeks to explain various kinds of moral phenomena wholly or partly in genetic terms. Ethical topics addressed include altruistic behaviors, conservation ethics, an innate sense of fairness, a capacity for normative guidance, feelings of kindness or love, self-sacrifice, incest-avoidance, parental care, in-group loyalty, monogamy, feelings related to competitiveness and retribution, moral "cheating," and hypocrisy.

A key issue in evolutionary psychology has been how altruistic feelings and behaviors could have evolved, in both humans and nonhumans, when the process of natural selection is based on the multiplication over time only of those genes that adapt better to changes in the environment of the species. Theories addressing this have included kin selection, group selection, and reciprocal altruism (both direct and indirect, and on a society-wide scale). Descriptive evolutionary ethicists have also debated whether various types of moral phenomena should be seen as adaptations which have evolved because of their direct adaptive benefits, or spin-offs that evolved as side-effects of adaptive behaviors.

Normative evolutionary ethics

Normative evolutionary ethics is the most controversial branch of evolutionary ethics. Normative evolutionary ethics aims at defining which acts are right or wrong, and which things are good or bad, in evolutionary terms. It is not merely describing, but it is prescribing goals, values and obligations. Social Darwinism, discussed above, is the most historically influential version of normative evolutionary ethics. As philosopher G. E. Moore famously argued, many early versions of normative evolutionary ethics seemed to commit a logical mistake that Moore dubbed the naturalistic fallacy. This was the mistake of defining a normative property, such as goodness, in terms of some non-normative, naturalistic property, such as pleasure or survival.

More sophisticated forms of normative evolutionary ethics need not commit either the naturalistic fallacy or the is-ought fallacy. But all varieties of normative evolutionary ethics face the difficult challenge of explaining how evolutionary facts can have normative authority for rational agents. "Regardless of why one has a given trait, the question for a rational agent is always: is it right for me to exercise it, or should I instead renounce and resist it as far as I am able?"

Evolutionary metaethics

Evolutionary theory may not be able to tell us what is morally right or wrong, but it might be able to illuminate our use of moral language, or to cast doubt on the existence of objective moral facts or the possibility of moral knowledge. Evolutionary ethicists such as Michael Ruse, E. O. Wilson, Richard Joyce, and Sharon Street have defended such claims.

Some philosophers who support evolutionary meta-ethics use it to undermine views of human well-being that rely upon Aristotelian teleology, or other goal-directed accounts of human flourishing. A number of thinkers have appealed to evolutionary theory in an attempt to debunk moral realism or support moral skepticism. Sharon Street is one prominent ethicist who argues that evolutionary psychology undercuts moral realism. According to Street, human moral decision-making is "thoroughly saturated" with evolutionary influences. Natural selection, she argues, would have rewarded moral dispositions that increased fitness, not ones that track moral truths, should they exist. It would be a remarkable and unlikely coincidence if "morally blind" ethical traits aimed solely at survival and reproduction aligned closely with independent moral truths. So we cannot be confident that our moral beliefs accurately track objective moral truth. Consequently, realism forces us to embrace moral skepticism. Such skepticism, Street claims, is implausible. So we should reject realism and instead embrace some antirealist view that allows for rationally justified moral beliefs.

Defenders of moral realism have offered two sorts of replies. One is to deny that evolved moral responses would likely diverge sharply from moral truth. According to David Copp, for example, evolution would favor moral responses that promote social peace, harmony, and cooperation. But such qualities are precisely those that lie at the core of any plausible theory of objective moral truth. So Street's alleged "dilemma"—deny evolution or embrace moral skepticism—is a false choice.

A second response to Street is to deny that morality is as "saturated" with evolutionary influences as Street claims. William Fitzpatrick, for instance, argues that "[e]ven if there is significant evolutionary influence on the content of many of our moral beliefs, it remains possible that many of our moral beliefs are arrived at partly (or in some cases wholly) through autonomous moral reflection and reasoning, just as with our mathematical, scientific and philosophical beliefs." The wide variability of moral codes, both across cultures and historical time periods, is difficult to explain if morality is as pervasively shaped by genetic factors as Street claims.

Another common argument evolutionary ethicists use to debunk moral realism is to claim that the success of evolutionary psychology in explaining human ethical responses makes the notion of moral truth "explanatorily superfluous." If we can fully explain, for example, why parents naturally love and care for their children in purely evolutionary terms, there is no need to invoke any "spooky" realist moral truths to do any explanatory work. Thus, for reasons of theoretical simplicity we should not posit the existence of such truths and, instead, should explain the widely held belief in objective moral truth as "an illusion fobbed off on us by our genes in order to get us to cooperate with one another (so that our genes survive)."

Combining Darwinism with moral realism does not lead to unacceptable results in epistemology. No two worlds, that are non-normatively identical, can differ normatively. The instantiation of normative properties is metaphysically possible in a world like ours. The phylogenetic adoption of moral sense does not deprive ethical norms of independent and objective truth-values. A parallel with general theoretical principles exists, which being unchangeable in themselves are discovered during an investigation. Ethical a priori cognition is vindicated to the extent to which other a priori knowledge is available. Scrutinizing similar situations, the developing mind pondered idealized models subject to definite laws. In social relation, mutually acceptable behavior was mastered. A cooperative solution in rivalry among competitors is presented by Nash equilibrium. This behavioral pattern is not conventional (metaphysically constructive) but represents an objective relation similar to that of force or momentum equilibrium in mechanics.

Eigenvalue perturbation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Eigenvalue_perturbation

In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system ${\displaystyle Ax=\lambda x}$ that is perturbed from one with known eigenvectors and eigenvalues ${\displaystyle A_0{x}_0={\lambda }_0{x}_0}$. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues ${\displaystyle x_{0i},{\lambda }_{0i},i=1,\dots n}$ 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. I'm this case the kinetic energy provides the mass matrix ${\displaystyle M}$, the potential strain energy provides the rigidity matrix ${\displaystyle K}$.

With both methods, the following system of differential equations or matrix differential equation is derived: ${\displaystyle M\ddot{x}+B\dot{x}+Kx=0}$ with the mass matrix ${\displaystyle M}$ , the damping matrix ${\displaystyle B}$ and the rigidity matrix ${\displaystyle K}$. If the damping effect is neglected, ${\displaystyle B=0}$, and a solution of the form ${\displaystyle x=e^{i\omega t}u}$ is assumed,${\displaystyle u}$ and ${\displaystyle {\omega }^2}$are obtained as solutions to the generalized eigenvalue problem ${\displaystyle -{\omega }^{2}Mu+Ku=0}$.

Setting of perturbation for a generalized eigenvalue problem

Suppose the solutions to the generalized eigenvalue problem are known to be

${\displaystyle {\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}.(0)}$

where ${\displaystyle {\mathbf{K}}_0}$ and ${\displaystyle {\mathbf{M}}_0}$ are matrices. That is, we know the eigenvalues λ_0i and eigenvectors x_0i for i = 1, ..., N. 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

${\displaystyle \mathbf{K}{\mathbf{x}}_i={\lambda }_i\mathbf{M}{\mathbf{x}}_i(1)}$

where

${\displaystyle \begin{array}{rl}\mathbf{K}& ={\mathbf{K}}_0+\delta \mathbf{K}\\ \mathbf{M}& ={\mathbf{M}}_0+\delta \mathbf{M}\end{array}}$

with the perturbations ${\displaystyle \delta \mathbf{K}}$ and ${\displaystyle \delta \mathbf{M}}$ much smaller than ${\displaystyle \mathbf{K}}$ and ${\displaystyle \mathbf{M}}$ respectively. Then the new eigenvalues and eigenvectors are expected to be similar to the original, plus small perturbations:

${\displaystyle \begin{array}{rl}{\lambda }_i& ={\lambda }_{0i}+\delta {\lambda }_i\\ {\mathbf{x}}_i& ={\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i\end{array}}$

Steps

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

${\displaystyle {\mathbf{x}}_{0j}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}={\delta }_{ij},}$${\displaystyle {\mathbf{x}}_i^T\mathbf{M}{\mathbf{x}}_j={\delta }_{ij}(2)}$

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

${\displaystyle \mathbf{K}{\mathbf{x}}_i-{\lambda }_i\mathbf{M}{\mathbf{x}}_i=0.}$

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

First order expansion of the equation

Substituting in (1) results in

${\displaystyle ({\mathbf{K}}_0+\delta \mathbf{K})({\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i)=\left({\lambda }_{0i}+\delta {\lambda }_i\right)\left({\mathbf{M}}_0+\delta \mathbf{M}\right)\left({\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i\right),}$

which expands to

${\displaystyle \begin{array}{rl}{\mathbf{K}}_0{\mathbf{x}}_{0i}& +\delta \mathbf{K}{\mathbf{x}}_{0i}+{\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}\delta {\mathbf{x}}_i=\\ & {\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}+\\ & {\lambda }_{0i}\delta \mathbf{M}\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}\delta {\mathbf{x}}_i.\end{array}}$

Canceling from (0) (${\displaystyle {\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}}$) leaves

${\displaystyle \begin{array}{rl}\delta \mathbf{K}{\mathbf{x}}_{0i}+& {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}\delta {\mathbf{x}}_i={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}+\\ & {\lambda }_{0i}\delta \mathbf{M}\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}\delta {\mathbf{x}}_i.\end{array}}$

Removing the higher-order terms, this simplifies to

${\displaystyle {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}.(3)}$

In other words, ${\displaystyle \delta {\lambda }_i}$ no longer denotes the exact variation of the eigenvalue but its first order approximation.

As the matrix is symmetric, the unperturbed eigenvectors are ${\displaystyle M}$ orthogonal and so can be used as a basis for the perturbed eigenvectors. This is the same as

${\displaystyle \delta {\mathbf{x}}_i=\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}(4)}$ with ${\displaystyle {\epsilon }_{ij}={\mathbf{x}}_{0j}^{T}M\delta {\mathbf{x}}_i}$,

where ε_ij are small constants that are to be determined.

In the same way, substituting in (2), and removing higher order terms, ${\displaystyle \delta {\mathbf{x}}_j{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\mathbf{x}}_{0j}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}\delta {\mathbf{M}}_0{\mathbf{x}}_{0i}=0(5)}$

The derivation is then split into two paths.

First path: get first eigenvalue perturbation

Eigenvalue perturbation

Starting with (3)${\displaystyle {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i};}$which is then left multiplied with ${\displaystyle {\mathbf{x}}_{0i}^T}$ along with using (2) as well as its first order variation (5); one gets

${\displaystyle {\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i}$

or

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}-{\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}$

This is the first order perturbation of the generalized Rayleigh quotient with fixed ${\displaystyle x_{0i}}$: ${\displaystyle R(K,M;x_{0i})=x_{0i}^{T}K{x}_{0i}/x_{0i}^{T}M{x}_{0i},\text{ with }{x}_{0i}^{T}M{x}_{0i}=1}$

Moreover, for ${\displaystyle M=I}$, the formula ${\displaystyle \delta {\lambda }_i=x_{0i}^T\delta K{x}_{0i}}$ should be compared with Bauer-Fike theorem which provides a bound for eigenvalue perturbation.

Eigenvector perturbation

One then left multiplies (3) with ${\displaystyle x_{0j}^T}$ for ${\displaystyle j\ne i}$ and get

${\displaystyle {\mathbf{x}}_{0j}^T{\mathbf{K}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0j}^T{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

Recalling that ${\displaystyle {\mathbf{x}}_{0j}^{T}K={\lambda }_{0j}{\mathbf{x}}_{0j}^{T}M\text{ and }{\mathbf{x}}_{0j}^T{\mathbf{M}}_0{\mathbf{x}}_{0i}=0,}$ for ${\displaystyle j\ne i}$, one may substitute for

${\displaystyle {\lambda }_{0j}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}.}$

or

${\displaystyle ({\lambda }_{0j}-{\lambda }_{0i}){\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}.}$

As the eigenvalues are assumed to be simple, for ${\displaystyle j\ne i}$

${\displaystyle {ϵ}_{ij}={\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i=\frac{-{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}{({\lambda }_{0j}-{\lambda }_{0i})},i=1,\dots N;j=1,\dots N;j\ne i.}$

Moreover (5) (the first order variation of (2) ) yields ${\displaystyle 2{ϵ}_{ii}=2{\mathbf{x}}_{0i}^T{\mathbf{M}}_0\delta x_i=-{\mathbf{x}}_{0i}^T\delta M{\mathbf{x}}_{0i}.}$ All of the components of ${\displaystyle \delta x_i}$ have now been obtained.

Second path: Straightforward manipulations

Substituting (4) into (3) and rearranging gives

${\displaystyle \begin{array}{rlrl}{\mathbf{K}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & (5)\\ \sum _{j=1}^N{\epsilon }_{ij}{\mathbf{K}}_0{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & \\ (\text{applying }{\mathbf{K}}_0\text{ to the sum})\\ \sum _{j=1}^N{\epsilon }_{ij}{\lambda }_{0j}{\mathbf{M}}_0{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & (\text{using Eq. }(1))\end{array}}$

Because the eigenvectors are M₀-orthogonal when M₀ is positive definite, one can remove the summations by left-multiplying by ${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}}$:

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\epsilon }_{ii}{\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

By use of equation (1) again:

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{K}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.(6)}$

The two terms containing ε_ii are equal because left-multiplying (1) by ${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}}$ gives

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

Canceling those terms in (6) leaves

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

Rearranging gives

${\displaystyle \delta {\lambda }_i=\frac{{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}}}$

But by (2), this denominator is equal to 1. Thus

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}.}$

Then, as ${\displaystyle {\lambda }_i\ne {\lambda }_k}$ for ${\displaystyle i\ne k}$ (this is the assumption of simple eigenvalues) by left-multiplying equation (5) by ${\displaystyle {\mathbf{x}}_{0k}^{\mathrm{\top }}}$:

${\displaystyle {\epsilon }_{ik}=\frac{{\mathbf{x}}_{0k}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0k}},i\ne k.}$

Or by changing the name of the indices:

${\displaystyle {\epsilon }_{ij}=\frac{{\mathbf{x}}_{0j}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0j}},i\ne j.}$

To find ε_ii, using the fact that

${\displaystyle {\mathbf{x}}_i^{\mathrm{\top }}\mathbf{M}{\mathbf{x}}_i=1}$

implies:

${\displaystyle {\epsilon }_{ii}=-\frac{1}{2}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}.}$

Summary of the first order perturbation result

In the case where all the matrices are Hermitian positive definite and all the eigenvalues are distinct,

${\displaystyle \begin{array}{rl}{\lambda }_i& ={\lambda }_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}\\ {\mathbf{x}}_i& ={\mathbf{x}}_{0i}\left(1-\frac{1}{2}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}\right)+\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N\frac{{\mathbf{x}}_{0j}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0j}}{\mathbf{x}}_{0j}\end{array}}$

for infinitesimal ${\displaystyle \delta \mathbf{K}}$ and ${\displaystyle \delta \mathbf{M}}$ (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 ${\displaystyle f:{\mathbb{R}}^{n+m}\to {\mathbb{R}}^m,f:(x,y)\mapsto f(x,y)}$, with an invertible Jacobian matrix ${\displaystyle J_{f,b}(x_0,y_0)}$, from a point ${\displaystyle (x_0,y_0)}$ solution of ${\displaystyle f(x_0,y_0)=0}$, we get solutions of ${\displaystyle f(x,y)=0}$ with ${\displaystyle x}$ close to ${\displaystyle x_0}$ in the form ${\displaystyle y=g(x)}$ where ${\displaystyle g}$ is a continuously differentiable function ; moreover the Jacobian marix of ${\displaystyle g}$ is provided by the linear system

${\displaystyle J_{f,y}(x,g(x))J_{g,x}(x)+J_{f,x}(x,g(x))=0(6)}$.

As soon as the hypothesis of the theorem is satisfied, the Jacobian matrix of ${\displaystyle g}$ may be computed with a first order expansion of ${\displaystyle f(x_0+\delta x,y_0+\delta y)=0}$, we get

${\displaystyle J_{f,x}(x,g(x))\delta x+J_{f,y}(x,g(x))\delta y=0}$; as ${\displaystyle \delta y=J_{g,x}(x)\delta x}$, it is equivalent to equation ${\displaystyle (6)}$.

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 ${\displaystyle \tilde{f}:{\mathbb{R}}^{2n^2}\times {\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}}$, with

• ${\displaystyle \tilde{f}(K,M,\lambda ,x)=(\genfrac{}{}{0ex}{}{f(K,M,\lambda ,x)}{f_{n+1}(x)})}$ with

${\displaystyle f(K,M,\lambda ,x)=Kx-\lambda x,f_{n+1}(M,x)=x^{T}Mx-1}$. In order to use the Implicit function theorem, we study the invertibility of the Jacobian ${\displaystyle J_{\tilde{f};\lambda ,x}(K,M;{\lambda }_{0i},x_{0i})}$ with

${\displaystyle J_{\tilde{f};\lambda ,x}(K,M;{\lambda }_i,x_i)(\delta \lambda ,\delta x)=(\genfrac{}{}{0ex}{}{-M{x}_i}{0})\delta \lambda +(\genfrac{}{}{0ex}{}{K-\lambda M}{2x_i^{T}M})\delta x_i}$. Indeed, the solution of

${\displaystyle J_{\tilde{f};{\lambda }_{0i},x_{0i}}(K,M;{\lambda }_{0i},x_{0i})(\delta {\lambda }_i,\delta x_i)=}$${\displaystyle (\genfrac{}{}{0ex}{}{y}{y_{n+1}})}$ may be derived with computations similar to the derivation of the expansion.

$${\displaystyle \delta {\lambda }_i=-x_{0i}^{T}y,\text{ and }({\lambda }_{0i}-{\lambda }_{0j})x_{0j}^{T}M\delta x_i=x_j^{T}y,j=1,\dots ,n,j\ne i;}$$ ${\displaystyle \text{ or }{x}_{0j}^{T}M\delta x_i=x_j^{T}y/({\lambda }_{0i}-{\lambda }_{0j}),\text{ and }2x_{0i}^{T}M\delta x_i=y_{n+1}}$

When ${\displaystyle {\lambda }_i}$ is a simple eigenvalue, as the eigenvectors ${\displaystyle x_{0j},j=1,\dots ,n}$ 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 ${\displaystyle (K,M)\mapsto ({\lambda }_i(K,M),x_i(K,M))}$ hence the expansion with little o notation: ${\displaystyle {\lambda }_i={\lambda }_{0i}+\delta {\lambda }_i+o(\Vert \delta K\Vert +\Vert \delta M\Vert )}$ ${\displaystyle x_i=x_{0i}+\delta x_i+o(\Vert \delta K\Vert +\Vert \delta M\Vert )}$. with

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}-{\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i};}$ ${\displaystyle \delta x_i={\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i{\mathbf{x}}_{0j}\text{ with}}$${\displaystyle {\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i=\frac{-{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}{({\lambda }_{0j}-{\lambda }_{0i})},i=1,\dots n;j=1,\dots n;j\ne i.}$ 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 λ_i as a function of changes in the entries of the matrices. (Recall that the matrices are symmetric and so changing K_kℓ will also change K_ℓk, hence the (2 − δ_kℓ) 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 ${\displaystyle M}$, the potential strain energy provides the rigidity matrix ${\displaystyle K}$. 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 ${\displaystyle M\ddot{x}+B\dot{x}+Kx=0}$ with the mass matrix ${\displaystyle M}$ , the damping matrix ${\displaystyle B}$ and the rigidity matrix ${\displaystyle K}$. If we neglect the damping effect, we use ${\displaystyle B=0}$, we can look for a solution of the following form ${\displaystyle x=e^{i\omega t}u}$; we obtain that ${\displaystyle u}$ and ${\displaystyle {\omega }^2}$are solution of the generalized eigenvalue problem ${\displaystyle -{\omega }^{2}Mu+Ku=0}$

Setting of perturbation for a generalized eigenvalue problem

Suppose we have solutions to the generalized eigenvalue problem,

${\displaystyle {\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}.(0)}$

where ${\displaystyle {\mathbf{K}}_0}$ and ${\displaystyle {\mathbf{M}}_0}$ are matrices. That is, we know the eigenvalues λ_0i and eigenvectors x_0i for i = 1, ..., N. 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

${\displaystyle \mathbf{K}{\mathbf{x}}_i={\lambda }_i\mathbf{M}{\mathbf{x}}_i(1)}$

where

${\displaystyle \begin{array}{rl}\mathbf{K}& ={\mathbf{K}}_0+\delta \mathbf{K}\\ \mathbf{M}& ={\mathbf{M}}_0+\delta \mathbf{M}\end{array}}$

with the perturbations ${\displaystyle \delta \mathbf{K}}$ and ${\displaystyle \delta \mathbf{M}}$ much smaller than ${\displaystyle \mathbf{K}}$ and ${\displaystyle \mathbf{M}}$ respectively. Then we expect the new eigenvalues and eigenvectors to be similar to the original, plus small perturbations:

${\displaystyle \begin{array}{rl}{\lambda }_i& ={\lambda }_{0i}+\delta {\lambda }_i\\ {\mathbf{x}}_i& ={\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i\end{array}}$

Steps

We assume that the matrices are symmetric and positive definite, and assume we have scaled the eigenvectors such that

${\displaystyle {\mathbf{x}}_{0j}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}={\delta }_{ij},}$${\displaystyle {\mathbf{x}}_i^T\mathbf{M}{\mathbf{x}}_j={\delta }_{ij}(2)}$

where δ_ij is the Kronecker delta. Now we want to solve the equation

${\displaystyle \mathbf{K}{\mathbf{x}}_i-{\lambda }_i\mathbf{M}{\mathbf{x}}_i=0.}$

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

First order expansion of the equation

Substituting in (1), we get

${\displaystyle ({\mathbf{K}}_0+\delta \mathbf{K})({\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i)=\left({\lambda }_{0i}+\delta {\lambda }_i\right)\left({\mathbf{M}}_0+\delta \mathbf{M}\right)\left({\mathbf{x}}_{0i}+\delta {\mathbf{x}}_i\right),}$

which expands to

${\displaystyle \begin{array}{rl}{\mathbf{K}}_0{\mathbf{x}}_{0i}& +\delta \mathbf{K}{\mathbf{x}}_{0i}+{\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}\delta {\mathbf{x}}_i=\\ & {\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}+\\ & {\lambda }_{0i}\delta \mathbf{M}\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}\delta {\mathbf{x}}_i.\end{array}}$

Canceling from (0) (${\displaystyle {\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}}$) leaves

${\displaystyle \begin{array}{rl}\delta \mathbf{K}{\mathbf{x}}_{0i}+& {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}\delta {\mathbf{x}}_i={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}+\\ & {\lambda }_{0i}\delta \mathbf{M}\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0\delta {\mathbf{x}}_i+\delta {\lambda }_i\delta \mathbf{M}\delta {\mathbf{x}}_i.\end{array}}$

Removing the higher-order terms, this simplifies to

${\displaystyle {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}.(3)}$

In other words, ${\displaystyle \delta {\lambda }_i}$ no longer denotes the exact variation of the eigenvalue but its first order approximation.

As the matrix is symmetric, the unperturbed eigenvectors are ${\displaystyle M}$ orthogonal and so we use them as a basis for the perturbed eigenvectors. That is, we want to construct

${\displaystyle \delta {\mathbf{x}}_i=\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}(4)}$ with ${\displaystyle {\epsilon }_{ij}={\mathbf{x}}_{0j}^{T}M\delta {\mathbf{x}}_i}$,

where the ε_ij are small constants that are to be determined.

In the same way, substituting in (2), and removing higher order terms, we get ${\displaystyle \delta {\mathbf{x}}_j{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\mathbf{x}}_{0j}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}\delta {\mathbf{M}}_0{\mathbf{x}}_{0i}=0(5)}$

The derivation can go on with two forks.

First fork: get first eigenvalue perturbation

Eigenvalue perturbation

We start with (3)${\displaystyle {\mathbf{K}}_0\delta {\mathbf{x}}_i+\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i};}$

we left multiply with ${\displaystyle {\mathbf{x}}_{0i}^T}$ and use (2) as well as its first order variation (5); we get

${\displaystyle {\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i}$

or

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}-{\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}$

We notice that it is the first order perturbation of the generalized Rayleigh quotient with fixed ${\displaystyle x_{0i}}$: ${\displaystyle R(K,M;x_{0i})=x_{0i}^{T}K{x}_{0i}/x_{0i}^{T}M{x}_{0i},\text{ with }{x}_{0i}^{T}M{x}_{0i}=1}$

Moreover, for ${\displaystyle M=I}$, the formula ${\displaystyle \delta {\lambda }_i=x_{0i}^T\delta K{x}_{0i}}$ should be compared with Bauer-Fike theorem which provides a bound for eigenvalue perturbation.

Eigenvector perturbation

We left multiply (3) with ${\displaystyle x_{0j}^T}$ for ${\displaystyle j\ne i}$ and get

${\displaystyle {\mathbf{x}}_{0j}^T{\mathbf{K}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0j}^T{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

We use ${\displaystyle {\mathbf{x}}_{0j}^{T}K={\lambda }_{0j}{\mathbf{x}}_{0j}^{T}M\text{ and }{\mathbf{x}}_{0j}^T{\mathbf{M}}_0{\mathbf{x}}_{0i}=0,}$ for ${\displaystyle j\ne i}$.

${\displaystyle {\lambda }_{0j}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}.}$

or

${\displaystyle ({\lambda }_{0j}-{\lambda }_{0i}){\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i+{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}.}$

As the eigenvalues are assumed to be simple, for ${\displaystyle j\ne i}$

${\displaystyle {ϵ}_{ij}={\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i=\frac{-{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}{({\lambda }_{0j}-{\lambda }_{0i})},i=1,\dots N;j=1,\dots N;j\ne i.}$

Moreover (5) (the first order variation of (2) ) yields ${\displaystyle 2{ϵ}_{ii}=2{\mathbf{x}}_{0i}^T{\mathbf{M}}_0\delta x_i=-{\mathbf{x}}_{0i}^T\delta M{\mathbf{x}}_{0i}.}$ We have obtained all the components of ${\displaystyle \delta x_i}$ .

Second fork: Straightforward manipulations

Substituting (4) into (3) and rearranging gives

${\displaystyle \begin{array}{rlrl}{\mathbf{K}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & (5)\\ \sum _{j=1}^N{\epsilon }_{ij}{\mathbf{K}}_0{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & \\ (\text{applying }{\mathbf{K}}_0\text{ to the sum})\\ \sum _{j=1}^N{\epsilon }_{ij}{\lambda }_{0j}{\mathbf{M}}_0{\mathbf{x}}_{0j}+\delta \mathbf{K}{\mathbf{x}}_{0i}& ={\lambda }_{0i}{\mathbf{M}}_0\sum _{j=1}^N{\epsilon }_{ij}{\mathbf{x}}_{0j}+{\lambda }_{0i}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{M}}_0{\mathbf{x}}_{0i}& & (\text{using Eq. }(1))\end{array}}$

Because the eigenvectors are M₀-orthogonal when M₀ is positive definite, we can remove the summations by left-multiplying by ${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}}$:

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\epsilon }_{ii}{\lambda }_{0i}{\mathbf{M}}_0{\mathbf{x}}_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

By use of equation (1) again:

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{K}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\epsilon }_{ii}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.(6)}$

The two terms containing ε_ii are equal because left-multiplying (1) by ${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}}$ gives

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{K}}_0{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

Canceling those terms in (6) leaves

${\displaystyle {\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{K}{\mathbf{x}}_{0i}={\lambda }_{0i}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}+\delta {\lambda }_i{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}.}$

Rearranging gives

${\displaystyle \delta {\lambda }_i=\frac{{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\mathbf{x}}_{0i}^{\mathrm{\top }}{\mathbf{M}}_0{\mathbf{x}}_{0i}}}$

But by (2), this denominator is equal to 1. Thus

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}.}$

Then, as ${\displaystyle {\lambda }_i\ne {\lambda }_k}$ for ${\displaystyle i\ne k}$ (assumption simple eigenvalues) by left-multiplying equation (5) by ${\displaystyle {\mathbf{x}}_{0k}^{\mathrm{\top }}}$:

${\displaystyle {\epsilon }_{ik}=\frac{{\mathbf{x}}_{0k}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0k}},i\ne k.}$

Or by changing the name of the indices:

${\displaystyle {\epsilon }_{ij}=\frac{{\mathbf{x}}_{0j}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0j}},i\ne j.}$

To find ε_ii, use the fact that:

${\displaystyle {\mathbf{x}}_i^{\mathrm{\top }}\mathbf{M}{\mathbf{x}}_i=1}$

implies:

${\displaystyle {\epsilon }_{ii}=-\frac{1}{2}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}.}$

Summary of the first order perturbation result

In the case where all the matrices are Hermitian positive definite and all the eigenvalues are distinct,

${\displaystyle \begin{array}{rl}{\lambda }_i& ={\lambda }_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}\\ {\mathbf{x}}_i& ={\mathbf{x}}_{0i}\left(1-\frac{1}{2}{\mathbf{x}}_{0i}^{\mathrm{\top }}\delta \mathbf{M}{\mathbf{x}}_{0i}\right)+\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N\frac{{\mathbf{x}}_{0j}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}}{{\lambda }_{0i}-{\lambda }_{0j}}{\mathbf{x}}_{0j}\end{array}}$

for infinitesimal ${\displaystyle \delta \mathbf{K}}$ and ${\displaystyle \delta \mathbf{M}}$ (the higher order terms in (3) being neglected).

So far, we have not proved that these higher order terms may be neglected. This point may be derived using the implicit function theorem; in next section, we summarize the use of this theorem in order to obtain a first order expansion.

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 ${\displaystyle f:{\mathbb{R}}^{n+m}\to {\mathbb{R}}^m,f:(x,y)\mapsto f(x,y)}$, with an invertible Jacobian matrix ${\displaystyle J_{f,b}(x_0,y_0)}$, from a point ${\displaystyle (x_0,y_0)}$ solution of ${\displaystyle f(x_0,y_0)=0}$, we get solutions of ${\displaystyle f(x,y)=0}$ with ${\displaystyle x}$ close to ${\displaystyle x_0}$ in the form ${\displaystyle y=g(x)}$ where ${\displaystyle g}$ is a continuously differentiable function ; moreover the Jacobian marix of ${\displaystyle g}$ is provided by the linear system

${\displaystyle J_{f,y}(x,g(x))J_{g,x}(x)+J_{f,x}(x,g(x))=0(6)}$.

As soon as the hypothesis of the theorem is satisfied, the Jacobian matrix of ${\displaystyle g}$ may be computed with a first order expansion of ${\displaystyle f(x_0+\delta x,y_0+\delta y)=0}$, we get

${\displaystyle J_{f,x}(x,g(x))\delta x+J_{f,y}(x,g(x))\delta y=0}$; as ${\displaystyle \delta y=J_{g,x}(x)\delta x}$, it is equivalent to equation ${\displaystyle (6)}$.

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 ${\displaystyle \tilde{f}:{\mathbb{R}}^{2n^2}\times {\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}}$, with

• ${\displaystyle \tilde{f}(K,M,\lambda ,x)=(\genfrac{}{}{0ex}{}{f(K,M,\lambda ,x)}{f_{n+1}(x)})}$ with

${\displaystyle f(K,M,\lambda ,x)=Kx-\lambda x,f_{n+1}(M,x)=x^{T}Mx-1}$. In order to use the Implicit function theorem, we study the invertibility of the Jacobian ${\displaystyle J_{\tilde{f};\lambda ,x}(K,M;{\lambda }_{0i},x_{0i})}$ with

${\displaystyle J_{\tilde{f};\lambda ,x}(K,M;{\lambda }_i,x_i)(\delta \lambda ,\delta x)=(\genfrac{}{}{0ex}{}{-M{x}_i}{0})\delta \lambda +(\genfrac{}{}{0ex}{}{K-\lambda M}{2x_i^{T}M})\delta x_i}$. Indeed, the solution of

${\displaystyle J_{\tilde{f};{\lambda }_{0i},x_{0i}}(K,M;{\lambda }_{0i},x_{0i})(\delta {\lambda }_i,\delta x_i)=}$${\displaystyle (\genfrac{}{}{0ex}{}{y}{y_{n+1}})}$ may be derived with computations similar to the derivation of the expansion.

$${\displaystyle \delta {\lambda }_i=-x_{0i}^{T}y,\text{ and }({\lambda }_{0i}-{\lambda }_{0j})x_{0j}^{T}M\delta x_i=x_j^{T}y,j=1,\dots ,n,j\ne i;}$$ ${\displaystyle \text{ or }{x}_{0j}^{T}M\delta x_i=x_j^{T}y/({\lambda }_{0i}-{\lambda }_{0j}),\text{ and }2x_{0i}^{T}M\delta x_i=y_{n+1}}$

When ${\displaystyle {\lambda }_i}$ is a simple eigenvalue, as the eigenvectors ${\displaystyle x_{0j},j=1,\dots ,n}$ 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 ${\displaystyle (K,M)\mapsto ({\lambda }_i(K,M),x_i(K,M))}$ hence the expansion with little o notation: ${\displaystyle {\lambda }_i={\lambda }_{0i}+\delta {\lambda }_i+o(\Vert \delta K\Vert +\Vert \delta M\Vert )}$ ${\displaystyle x_i=x_{0i}+\delta x_i+o(\Vert \delta K\Vert +\Vert \delta M\Vert )}$. with

${\displaystyle \delta {\lambda }_i={\mathbf{x}}_{0i}^T\delta \mathbf{K}{\mathbf{x}}_{0i}-{\lambda }_{0i}{\mathbf{x}}_{0i}^T\delta \mathbf{M}{\mathrm{x}}_{0i};}$ ${\displaystyle \delta x_i={\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i{\mathbf{x}}_{0j}\text{ with}}$${\displaystyle {\mathbf{x}}_{0j}^T{\mathbf{M}}_0\delta {\mathbf{x}}_i=\frac{-{\mathbf{x}}_{0j}^T\delta \mathbf{K}{\mathbf{x}}_{0i}+{\lambda }_{0i}{\mathbf{x}}_{0j}^T\delta \mathbf{M}{\mathrm{x}}_{0i}}{({\lambda }_{0j}-{\lambda }_{0i})},i=1,\dots n;j=1,\dots n;j\ne i.}$ 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 λ_i as a function of changes in the entries of the matrices. (Recall that the matrices are symmetric and so changing K_kℓ will also change K_ℓk, hence the (2 − δ_kℓ) term.)

${\displaystyle \begin{array}{rl}\frac{\mathrm{\partial }{\lambda }_i}{\mathrm{\partial }{\mathbf{K}}_{(k\ell )}}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathbf{K}}_{(k\ell )}}\left({\lambda }_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}\right)=x_{0i(k)}{x}_{0i(\ell )}\left(2-{\delta }_{k\ell }\right)\\ \frac{\mathrm{\partial }{\lambda }_i}{\mathrm{\partial }{\mathbf{M}}_{(k\ell )}}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathbf{M}}_{(k\ell )}}\left({\lambda }_{0i}+{\mathbf{x}}_{0i}^{\mathrm{\top }}\left(\delta \mathbf{K}-{\lambda }_{0i}\delta \mathbf{M}\right){\mathbf{x}}_{0i}\right)=-{\lambda }_i{x}_{0i(k)}{x}_{0i(\ell )}\left(2-{\delta }_{k\ell }\right).\end{array}}$

Similarly

${\displaystyle \begin{array}{rl}\frac{\mathrm{\partial }{\mathbf{x}}_i}{\mathrm{\partial }{\mathbf{K}}_{(k\ell )}}& =\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N\frac{x_{0j(k)}{x}_{0i(\ell )}\left(2-{\delta }_{k\ell }\right)}{{\lambda }_{0i}-{\lambda }_{0j}}{\mathbf{x}}_{0j}\\ \frac{\mathrm{\partial }{\mathbf{x}}_i}{\mathrm{\partial }{\mathbf{M}}_{(k\ell )}}& =-{\mathbf{x}}_{0i}\frac{x_{0i(k)}{x}_{0i(\ell )}}{2}(2-{\delta }_{k\ell })-\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N\frac{{\lambda }_{0i}{x}_{0j(k)}{x}_{0i(\ell )}}{{\lambda }_{0i}-{\lambda }_{0j}}{\mathbf{x}}_{0j}\left(2-{\delta }_{k\ell }\right).\end{array}}$

Eigenvalue sensitivity, a small example

A simple case is ${\displaystyle K=\left[\begin{array}{cc}2& b\\ b& 0\end{array}\right]}$; however you can compute eigenvalues and eigenvectors with the help of online tools such as (see introduction in Wikipedia WIMS) or using Sage SageMath. You get the smallest eigenvalue ${\displaystyle \lambda =-\left[\sqrt{b^2+1}+1\right]}$ and an explicit computation ${\displaystyle \frac{\mathrm{\partial }\lambda }{\mathrm{\partial }b}=\frac{-x}{\sqrt{x^2+1}}}$; more over, an associated eigenvector is ${\displaystyle {\tilde{x}}_0=[x,-(\sqrt{x^2+1}+1)){]}^T}$; it is not an unitary vector; so ${\displaystyle x_{01}{x}_{02}={\tilde{x}}_{01}{\tilde{x}}_{02}/\Vert {\tilde{x}}_0{\Vert }^2}$; we get ${\displaystyle \Vert {\tilde{x}}_0{\Vert }^2=2\sqrt{x^2+1}(\sqrt{x^2+1}+1)}$ and ${\displaystyle {\tilde{x}}_{01}{\tilde{x}}_{02}=-x(\sqrt{x^2+1}+1)}$ ; hence ${\displaystyle x_{01}{x}_{02}=-\frac{x}{2\sqrt{x^2+1}}}$; for this example , we have checked that ${\displaystyle \frac{\mathrm{\partial }\lambda }{\mathrm{\partial }b}=2x_{01}{x}_{02}}$ or ${\displaystyle \delta \lambda =2x_{01}{x}_{02}\delta b}$.

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 ${\displaystyle N}$ linearly independent eigenvectors. An eigenvalue problem involving non-symmetric matrices is not guaranteed to have ${\displaystyle N}$ linearly independent eigenvectors, though a sufficient condition is that ${\displaystyle \mathbf{K}}$ and ${\displaystyle \mathbf{M}}$ be simultaneously diagonalizable.

The case of repeated eigenvalues

A technical report of Rellich  for perturbation of eigenvalue problems provides several examples. The elementary examples are in chapter 2. The report may be downloaded from archive.org. We draw an example in which the eigenvectors have a nasty behavior.

Example 1

Consider the following matrix ${\displaystyle B(ϵ)=ϵ\left[\begin{array}{cc}\cos (2/ϵ)& ,\sin (2/ϵ)\\ \sin (2/ϵ)& ,s\cos (2/ϵ)\end{array}\right]}$ and ${\displaystyle A(ϵ)=I-e^{-1/{ϵ}^2}B;}$ ${\displaystyle A(0)=I.}$ For ${\displaystyle ϵ\ne 0}$, the matrix ${\displaystyle A(ϵ)}$ has eigenvectors ${\displaystyle {\mathrm{\Phi }}^1=[\cos (1/ϵ),-\sin (1/ϵ){]}^T;{\mathrm{\Phi }}^2=[\sin (1/ϵ),-\cos (1/ϵ){]}^T}$ belonging to eigenvalues ${\displaystyle {\lambda }_1=1-e^{-1/{ϵ}^2)},{\lambda }_2=1+e^{-1/{ϵ}^2)}}$. Since ${\displaystyle {\lambda }_1\ne {\lambda }_2}$ for ${\displaystyle ϵ\ne 0}$ if ${\displaystyle u^j(ϵ),j=1,2,}$ are any normalized eigenvectors belonging to ${\displaystyle {\lambda }_j(ϵ),j=1,2}$ respectively then ${\displaystyle u^j=e^{{\alpha }_j(ϵ)}{\mathrm{\Phi }}^j(ϵ)}$ where ${\displaystyle {\alpha }_j,j=1,2}$ are real for ${\displaystyle ϵ\ne 0.}$ It is obviously impossible to define ${\displaystyle {\alpha }_1(ϵ)}$ , say, in such a way that ${\displaystyle u^1(ϵ)}$ tends to a limit as ${\displaystyle ϵ\to 0,}$ because ${\displaystyle |u^1(ϵ)|=|\cos (1/ϵ)|}$ has no limit as ${\displaystyle ϵ\to 0.}$

Note in this example that ${\displaystyle A_{jk}(ϵ)}$ is not only continuous but also has continuous derivatives of all orders. Rellich draws the following important consequence. << Since in general the individual eigenvectors do not depend continuously on the perturbation parameter even though the operator ${\displaystyle A(ϵ)}$ does, it is necessary to work, not with an eigenvector, but rather with the space spanned by all the eigenvectors belonging to the same eigenvalue. >>

Example 2

This example is less nasty that the previous one. Suppose ${\displaystyle [K_0]}$ is the 2x2 identity matrix, any vector is an eigenvector; then ${\displaystyle u_0=[1,1{]}^T/\sqrt{2}}$ is one possible eigenvector. But if one makes a small perturbation, such as

${\displaystyle [K]=[K_0]+\left[\begin{array}{cc}ϵ& 0\\ 0& 0\end{array}\right]}$

Then the eigenvectors are ${\displaystyle v_1=[1,0{]}^T}$ and ${\displaystyle v_2=[0,1{]}^T}$; they are constant with respect to ${\displaystyle ϵ}$ so that ${\displaystyle \Vert u_0-v_1\Vert }$ is constant and does not go to zero.