**3. The quadratic eigenvalue problems**

In practice, nonlinear eigenproblems commonly arise in dynamic/stability analysis of struc‐ tures 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 **Q**(*λ*)**x** = **0** where

$$\mathbf{Q}\{\lambda\} \colon = \lambda^2 A + \lambda B + \mathbb{C}, \quad A, \ B, \ \mathbb{C} \in \mathbb{C}^{n \times n}, \quad A \neq \mathbf{0}, \ \mathbf{x} \neq \mathbf{0} \tag{10}$$

This section is organized as follows:

**1.** Basic Properties

**Example 4.5.2** Let in GEP *A*=(

72 Applied Linear Algebra in Action

1 2

3 4), *<sup>B</sup>* =( <sup>1</sup> <sup>−</sup><sup>1</sup>

*pn*(*λ*) = (1 − *λ*)(4 − *λ*) − (3 + *λ*)(2 + *λ*) = − 2 − 10*λ* and the only eigenvalue value is *<sup>λ</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

and *B* is positive definite, that is, all the eigenvalues of the matrix *B* are positive.

positive definite, *B* has a Cholesky decomposition, i.e., *B* = *CC<sup>H</sup>*. Then is

l

orthonormal eigenvectors of F then for **x***<sup>i</sup>*

*ij*

d

on the basis of (7) are the same eigenvalues.

Rayleigh quotient as follows *R*(*A*,*B*)

**3. The quadratic eigenvalue problems**

is the quadratic problem **Q**(*λ*)**x** = **0** where

Let **y***<sup>i</sup>* i **y***<sup>j</sup>*

Atypical nature of we met in the last two examples are the result of the fact that in their matrix *B* was not regular. Therefore, it is usual in the GEP taken that *A* and *B* are Hermitian matrix

Our goal is to find a connection between this taken GEP and symmetric SEP. As we said *B* is

: := Û = -

Since matrix *F* has n eigenvalues and it belongs to the GEP also has n real eigenvalues. Namely

*H H i j Hi j*

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 **x**, **y***B* := **x***<sup>H</sup>By*. Now we're going to GEP (7) redefine

Rayleigh quotient applies to all theorems of section 2.3 with appropriate modification of the

In practice, nonlinear eigenproblems commonly arise in dynamic/stability analysis of struc‐ tures 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

*C C*

**yy x x**

() ( )

(*x*): = *<sup>x</sup> <sup>H</sup> Ax*

definition changes required. This gives self-generation Hermitian (symmetrical) SEP.

*H H j j*

*CC Bx*

( ) ( )

**x xx** *i i*

= =

= =

H

**-1 x= x yy x** *H H A B Fy C AC , C* (8)

= *C*<sup>−</sup>**Hy***<sup>i</sup>* **i x***<sup>j</sup>*

H

= *C*<sup>−</sup>**Hy***<sup>j</sup>*

*<sup>x</sup> <sup>H</sup> Bx* . With this predefined inneren product and

apply

<sup>−</sup>1 1 ). When we get to the characteristic polynomial

5 .

(9)

We consider Rayleigh functional and Minmax Characterization

**2.** Linearization

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.

**3.** Physical Background

We study vibration analysis of structural systems

#### **3.1. Basic Properties**

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. Varia‐ tional characterization is generalization of well known minmax characterization for the linear eigenvalue problems.

We consider nonlinear eigenvalue problems

$$T\left(\lambda\right)x = \mathbf{0},\tag{11}$$

where *T*(*λ*) ∈ ℂ*<sup>n</sup>* × *<sup>n</sup>*, *λ* ∈ *J* is a family of the Hermitian matrices depending continuously on the parameter *λ* ∈ *J* and *J* is a real open interval which may be unbounded.

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 **x** ∈ ℂ*<sup>n</sup>*, **x** ≠ **0** the scalar real equation

$$f\left(\boldsymbol{\lambda} ; \mathbf{x}\right) \coloneqq \mathbf{x}^{\mathbf{H}} \mathcal{T}\left(\boldsymbol{\lambda}\right) \mathbf{x} \tag{12}$$

has at most one solution *p*(*x*) ∈ *J*. Then *f*(*λ*; **x**) = 0 implicitly defines a functional *p* on some subset *D* ⊂ ℂ*<sup>n</sup>* which is called the Rayleigh functional of (11).

(B) for every *x* ∈ *D* and every *λ* ∈ *J* with *λ* ≠ *p*(*x*) it holds that (*λ* − *p*(*x*))*f*(*λ*; *x*) > 0.

If *p* is defined on *D* = ℂ*<sup>n</sup>*\{**0**} then the problem (1) is called **overdamped**, otherwise it is called **nonoverdamped**.

Generalizations of the minmax and the maxmin characterizations of the eigenvalues were proved by Duffin [6] for the quadratic case and by Rogers [7] 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 [8], which gives more information about the following minmax charac‐ terization for eigenvalues.
