Paragraph (d) in the previous theorem is known as Rayleigh principle. However it is numerically worthless, because for the determination example, we need eigenvector or
The following formulation is known as min max principle of Courant-Fischer and often favorable to use.
From the above it is clear that these theorems are important for the localization of eigenvalues.
The following is an algorithm that is in linear algebra known as Rayleigh quotient iteration and it reads as follows
Finally point out that the most effective for symmetric matrices Divide-and-Conquer method. This method was introduced by Cupen  and the first effective implementation is the work of Cu and Eisenstat . About this method, more information can be found in .
We have already mentioned that the problem of eigenvalues has numerous applications in engineering. Even more motivation to consider problems of eigenvalues, comes from their large application in technical disciplines. On a simple example of a mass-spring system illustrated with application of eigenvalues in engineering. We are assuming, that each spring has the same natural length
Applying Newton’s second law we get the following system
We are aware of vibration theory that
If the last equation twice differentiate by t, we get
If the last two equations obtained expressions for replace the initial system and write the system in matrix form, then we get
Equation (6) represents unsymmetrical eigenvalue problem.
More information on this case can be found in 
In this section we will deal with general linear eigenvalue problem or the problem
A common acronym for general linear eigenvalue problem is GEP. Now eigenvalue problems previously discussed is called the standard eigenvalue problem and tagging with SEP. In practice, the more often we meet with GEP than SEP. Now let’s consider some features of GEP and establish its relationship with SEP.
It is obvious that the eigenvalues of (7) zero of the characteristic polynomial, which is defined as
The following two examples illustrate this situation with GEP
Atypical nature of we met in the last two examples are the result of the fact that in their matrix
Our goal is to find a connection between this taken GEP and symmetric SEP. As we said
From equation (8) it is clear that the eigenvectors GEP (6) are an orthonormal vectors in relation to the new inner product is defined as 〈
In practice, nonlinear eigenproblems commonly arise in dynamic/stability analysis of structures and in fluid mechanics, electronic behavior of semiconductor hetero-structures, vibration of fluid–solid structures, vibration of sandwich plates, accelerator design, vibro-acoustics of piezoelectric/poroelestic structures, nonlinear integrated optics, regularization on total least squares problems and stability of delay differential equations. In practice, the most important is the quadratic problem
This section is organized as follows:
We consider Rayleigh functional and Minmax Characterization
A standard approach for investigating or numerically solving quadratic eigenvalue linearization problems, where the original problem is transformed into a generalized linear eigenvalues problem with the same spectrum.
We study vibration analysis of structural systems
Variational characterization is important for finding eigenvalues. In this section we give a brief review of variational characterization of nonlinear eigenvalue problems. Since the quadratic eigenproblems are a special case of nonlinear eigenvalue problems, results for nonlinear eigenvalue problems can be specially applied for the quadratic eigenvalue problems. Variational characterization is generalization of well known minmax characterization for the linear eigenvalue problems.
We consider nonlinear eigenvalue problems
Problems of this type arise in damped vibrations of structures, conservative gyroscopic systems, lateral buckling problems, problems with retarded arguments, fluid–solid vibrations, and quantum dot heterostructures.
To generalize the variational characterization of eigenvalues we need a generalization of the Rayleigh quotient. To this end we assume that
(A) for every fixed
has at most one solution
(B) for every
Generalizations of the minmax and the maxmin characterizations of the eigenvalues were proved by Duffin  for the quadratic case and by Rogers  for the general overdamped problems. For the nonoverdamped eigenproblems the natural ordering to call the smallest eigenvalue the first one, the second smallest the second one, etc., is not appropriate. The next theorem is proved in , which gives more information about the following minmax characterization for eigenvalues.
If there exists the
The minimum in (3) is attained for the invariant subspace of
Sylvester’s law of inertia has an important role in the nonlinear eigenvalue problems. We will briefly look back to the Sylvester’s law of inertia. With this purpose we define the inertia of the Hermitian matrix
Next, we consider a case that an extreme eigenvalue
We consider the quadratic eigenvalue problem in the label QEP .That is why we adapt to the real scalar equation (11) QEPu. In this way, we get
Natural candidates for the Rayleigh functionals of QEPa (
The Rayleigh functionals are the generalization of the Rayleigh quotient.
In this section we deal with the hyperbolic quadratic pencil as an example overdamped problems and gyroscopically stabilized pencil as an example of the changes that are not.
Now, let us look briefly the hyperbolic quadratic pencil. It is overdamped square pencil given by (10) in which the
Now we will look at gyroscopically stabilized system in the label GSS. A quadratic polynomial matrix
is gyroscopically stabilized if for some
where denotes the positive square root of
A eigenvalue λ is positive type if applies
Theorem (Barkwell, Lancaster, Markus 1992)
The spectrum of a gyroscopic stabilized pencil is real, i.e. Q is quasi-hyperbolic.
All eigenvalues are either of positive type or of negative type
Without loss of generality we will observe only positive eigenvalues value
Let now and functionals appropriate for GSS. With them, we can define the Rayleigh functionals i
Voss and Kostić are defined for this function given interval in which the eigenvalues can minmax characterize.
In order to minmax characterized all eigenvalues, we will introduce new Rayleigh functional. It is a new strategy. With this aim we matrices
Because of the conditions (20) are
We will observe the following problem inherent value
Applies following theorem
Theorem is proved.
Analogous to the (14) we defined the following functions
In the following theorem we give information about the properties
For each vector
For λ>0 a function
We have already mentioned that
Of (a) and because (21) and (22) follows that the function
Theorem is proved.
Define a new functional
Theorem is proved.
In this section we will deal with linearization. As mentioned linearization is standard procedure for reducing QEP on GEP with a view to facilitate the computation of eigenvalues. We have already seen that the problem of eigenvalues usually come as a result of solving differential equations or systems of differential equations. That is the basic idea of linearization came in the field of differential equations where the order of the differential equation of the second order can be lowered by introducing a system of two partial differential equations of the first order with two unknown functions.
The basic idea of linearization in QEPa is the introduction of shift
Then we get
The resulting equations are GEP because they can be written respectively in the form of
Since the corresponding GEPs all matrices 2
We look now at the application of eigenvalues of quadratic problems in engineering. The largest review of applications QEP is in the .We have already mentioned in the introduction to eigenvalue problem arises in connection with differential equations or systems of differential equations. In structural mechanics the most commonly are used differential equations and therefore the problem of eigenvalues. Note that the ultimate goal is to determine the effect of vibrations on the performance and reliability of the system, and to control these effects.
We will now demonstrate the linearization of QEP on a concrete example from the engineering. Low vibration system on
The introduction of shift
Now we’re going to QED (24) apply linearization method presented in section 3.2. Thus we have the appropriate GEP
When the system is undamped (
Because the most common matrix
Because of the great practical application eigenvalue problem occupies an important place in linear algebra. In this chapter, we discussed the linear and quadratic eigenvalues. In particular, an emphasis is on numerical methods such as the QR algorithm, Rayleigh quotient iteration for linear problems of eigenvalues and linearization and minmax characterization of quadratic problems eigenvalues. The whole chapter shows that the structure of the matrix, participating in the problem of eigenvalues, strongly influence the choice of the method itself. It is also clear that using the features of the structure matrix can do much more effectively existing algorithms. Thus, further studies are going to increase of use feature matrix involved in the problem of eigenvalues, with the aim of improving the effectiveness of the method. Finally, we point out that in this chapter we introduced new Rayleigh functionals for gyroscopically stabilized system that enables complete minmax (maxmin) characterization of eigenvalues. It’s a new strategy. We have proved all relevant features of new Rayleigh functionals.
© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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