**2.4. Physical background**

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 *l* and the same spring constant *k.* Finally, we can assume that the displacement of each spring in measured to its own local coordinate system with an origin at the spring's equilibrium position.

Applying Newton's second law we get the following system

$$m\_{\mathbf{l}} \frac{d^2 \mathbf{x}\_{\mathbf{l}}}{dt^2} - k \left(-2\mathbf{x}\_{\mathbf{l}} + \mathbf{x}\_2\right) = \mathbf{0},$$

$$m\_2 \frac{d^2 \mathbf{x}\_2}{dt^2} - k \left(\mathbf{x}\_{\mathbf{l}} - 2\mathbf{x}\_2\right) = \mathbf{0}.$$

We are aware of vibration theory that

$$\mathbf{x}\_l = a\_l \sin \alpha \mathbf{\mathfrak{x}},$$

where *ai* is the amplitude of the vibration of mass *i* and *ω* is the frequency of the vibration. If the last equation twice differentiate by t, we get

$$\frac{d^2\chi\_2}{dt^2} = -a\_l \text{oo}^2 \sin \alpha t.$$

If the last two equations obtained expressions for *xi* i *<sup>d</sup>* <sup>2</sup>*xi d t* <sup>2</sup> (*<sup>i</sup>* =1, 2) replace the initial system and write the system in matrix form, then we get

#### Eigenvalue Problems http://dx.doi.org/10.5772/62267 71

$$A\begin{pmatrix} a\_1 \\ a\_2 \end{pmatrix} = o^2 \begin{pmatrix} a\_1 \\ a\_2 \end{pmatrix},\tag{6}$$

where

**Theorem2.3.6.** Rayleigh quotient iteration converges cubic.

Applying Newton's second law we get the following system

2 1 1 1 2 2

*dt*

*d x m k xx*

*d x m kx x*

2 2 2 12 2

*dt*

2

*d x*

*dt* = -

**2.4. Physical background**

70 Applied Linear Algebra in Action

the spring's equilibrium position.

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 *xi* i

write the system in matrix form, then we get

where *ai*

Finally point out that the most effective for symmetric matrices Divide-and-Conquer method. This method was introduced by Cupen [3] and the first effective implementation is the work

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 *l* and the same spring constant *k.* Finally, we can assume that the displacement of each spring in measured to its own local coordinate system with an origin at

( )


2 0

2 0.

( )


sin , *i i xa t* = w

2 2 <sup>2</sup> ω sin . *i*

is the amplitude of the vibration of mass *i* and *ω* is the frequency of the vibration.

*a t*

w

*<sup>d</sup>* <sup>2</sup>*xi*

*d t* <sup>2</sup> (*<sup>i</sup>* =1, 2) replace the initial system and

of Cu and Eisenstat [4]. About this method, more information can be found in [4].

$$A = \begin{pmatrix} 2k & -\frac{k}{m\_1} \\ -\frac{k}{m\_2} & \frac{2k}{m\_2} \end{pmatrix}$$

Equation (6) represents unsymmetrical eigenvalue problem.

More information on this case can be found in [5]

#### **2.5. General Linear Eigenvalue Problem**

In this section we will deal with general linear eigenvalue problem or the problem

$$\mathbf{A}\mathbf{x} = \mathbf{B}\mathbf{x}, \quad \mathbf{x} \neq \mathbf{0},\tag{7}$$

where *A*, *B* ∈ *C*(*n*,*n*)

The scalar *λ* is called an eigenvalue of the problem (7)*,* and **x** said to be an eigenvector of (7) corresponding to *λ.*

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 *pn*(*λ*) := *det*(*A* − *λB*). In the case of GES degree polynomial *pn* is less than or equal to n. The characteristic polynomial of degree n has *pn* if and only if *B* is regular matrix. In case *B=I* we get SEP and in this case SEP has n eigenvalues. GEP and can be less than n eigenvalues. It can also happen that at GEP is *pn*(*λ*)≡0 and it that case GEP has infinitely many eigenvalues.

The following two examples illustrate this situation with GEP

**Example 4.5.1** Let in GEP *A*=( 0 0 0 0), *<sup>B</sup>* =( 0 6 0 0). Then worth *pn*(*λ*)≡0 and every *λ*∈ ℂ is eigenvalue i **x**=( 1 0 ) is the corresponding eigenvector.

**Example 4.5.2** Let in GEP *A*=( 1 2 3 4), *<sup>B</sup>* =( <sup>1</sup> <sup>−</sup><sup>1</sup> <sup>−</sup>1 1 ). When we get to the characteristic polynomial *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> 5 .

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

Our goal is to find a connection between this taken GEP and symmetric SEP. As we said *B* is positive definite, *B* has a Cholesky decomposition, i.e., *B* = *CC<sup>H</sup>*. Then is

$$A\mathbf{x} = \lambda B\mathbf{x} \Leftrightarrow F\mathbf{y} \coloneqq \mathbf{C}^{-1} A\mathbf{C}^{-H} \mathbf{y}, \quad \mathbf{y} \coloneqq \mathbf{C}^{H} \mathbf{x} \tag{8}$$

Since matrix *F* has n eigenvalues and it belongs to the GEP also has n real eigenvalues. Namely on the basis of (7) are the same eigenvalues.

Let **y***<sup>i</sup>* i **y***<sup>j</sup>* orthonormal eigenvectors of F then for **x***<sup>i</sup>* = *C*<sup>−</sup>**Hy***<sup>i</sup>* **i x***<sup>j</sup>* = *C*<sup>−</sup>**Hy***<sup>j</sup>* apply

$$\begin{aligned} \boldsymbol{\delta}\_{ij} &= \left(\mathbf{y}^{i}\right)^{H} \mathbf{y}^{j} = \left(\mathbf{C}^{H} \ \mathbf{x}^{i}\right)^{H} \mathbf{C}^{H} \mathbf{x}^{j} \\ &= \left(\mathbf{x}^{i}\right)^{H} \mathbf{C} \mathbf{C}^{H} \mathbf{x}^{j} = \left(\mathbf{x}^{i}\right)^{H} \mathbf{B} \mathbf{x}^{j} \end{aligned} \tag{9}$$

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 as follows *R*(*A*,*B*) (*x*): = *<sup>x</sup> <sup>H</sup> Ax <sup>x</sup> <sup>H</sup> Bx* . With this predefined inneren product and Rayleigh quotient applies to all theorems of section 2.3 with appropriate modification of the definition changes required. This gives self-generation Hermitian (symmetrical) SEP.
