Sobre las capacidades territoriales: LNSM y PNSM

( 5 .1) can be read as "

[a]

operates on

[ u]

to yield

'A

times

[ u]

," and Eq. (5.1) can be generalized to any mathematical operation as:

Lu = 'Au

_(5.2)

where

L

_{is an operator that can represent multiplication by a matrix, dif­}

ferentiation, integration, and so on,

u

_{is a vector or function, and}

'A

_{is a}

scalar constant. For example, if

L

_{represents second differentiation with}

respect to x, y is a function of x, and k is a constant, then Eq. (5.2) can have the form:

(5.3) Equation (5.2) is a general statement of an eigenvalue problem, where

'A

_{is called the eigenvalue associated with the operator}

L,

_and

u

_is

1. The word eigenvalue is derived from the German word eigenwert, which means "proper or characteristic value."

the

eigenvector

or

eigenfunction

corresponding to the eigenvalue 'A and the operator L.

Eigenvalues and eigenvectors arise in numerical methods and have special importance in science and engineering. For example, in the study of vibrations, the eigenvalues represent the natural frequencies of a system or component, and the eigenvectors represent the modes of these vibrations. It is important to identify these natural frequencies because when the system or component is subjected to periodic external loads (forces) at or near these frequencies, resonance can cause the response (motion) of the structure to be amplified, potentially leading to failure of the component. In mechanics of materials, the principal stresses are the eigenvalues of the stress matrix, and the principal direc­ tions are the directions of the associated eigenvectors. In quantum mechanics, eigenvalues are especially important. In Heisenberg's for­ mulation of quantum mechanics, there exists an operator L correspond­ ing to every observable quantity (i.e., any quantity that can be measured or inferred experimentally such as position, velocity, or energy). This operator L operates on an operand '¥ called the wave function, and if the result is proportional to the wave function-, if L'I' = c'I' ,-then the

value of the observable, c, is the eigenvalue and is said to be certain (i.e., can be known very precisely). In other words, the eigenvalues c corresponding to the observable are those values of the observable that have a nonzero probability of occurring (and therefore being observed). Examples of such operators from quantum mechanics are

ih

B'I' = E'I', where

ih .£( )

is the energy operator andE is the energy;

27t Bt 21tBt

h-7 -7 h h-7 . h _d7'

-i-V'I'

= p'I', w ere

-i-V( )

is t e momentum operator an p is

27t 27t

the linear momentum, where

i

=

H

and his Planck's constant. The

eigenvectors, also known as eigenstates, represent one of many states in which an object or a system may exist corresponding to a particular eigenvalue.

There is a link between eigenvalue problems involving differential equations and eigenvalue problems involving matrices

(5.1),

which are the focus in this section. Numerical solution of eigenvalue problems involving ordinary differential equations (ODEs) results in systems of simultaneous equations of the form

(5.1).

In other words, numerical determination of the eigenvalues in a problem involving an ODE reduces to finding the eigenvalues of an associated matrix

[a]

, resulting in a problem of the form

(5.1).

Beyond the physical importance of eigenvalues in science and engi­ neering, the eigenvalues of a matrix can also provide useful information about its properties in numerical calculations involving that matrix. Section 4. 7 showed that the Jacobi and Gauss-Seidel iterative methods can be written in the form of:

x(k+ l) l

=

b'. - [a] I x(k) l

It turns out that whether or not these iterative methods converge to a solution depends on the eigenvalues of the matrix [a]. Moreover, how quickly the iterations converge depends on the magnitudes of the eigen­ values of [a].

5.2 THE CHARACTERISTIC EQUATION

Determination of the eigenvalues of a matrix from Eq.

( 5 .1)

_{is accom­} plished by rewriting it in the form:

[a-AI ][u]

= 0

(5 .4)

where [/] is the identity matrix with the same dimensions as [a]. When written in this homogeneous form, it can be seen that if the matrix [a -AI] is nonsingular (i.e., if it has an inverse), then multiplying both sides of Eq.

(5 .4)

_{by [a -AI r1 yields the trivial solution [u]}

= 0 .

_On

the other hand, if [a -'AI ] is singular, that is, if it does not have an inverse, then a nontrivial solution for [u] is possible. Another way of stating this criterion is based on Cramer's rule (see Chapter

2):

_the

matrix [a -AI ] is singular if its determinant is zero:

det[a-'AJ]

= 0

(5 .5)

Equation

(5 .5)

_{is called the characteristic equation. For a given matrix}

[a], it yields a polynomial equation for 'A, whose roots are the eigen­ values. Once the eigenvalues are known, the eigenvectors can be deter­ mined. This is done by substituting the eigenvalues (one at a time) in Eq.

(5 .1)

_{and solving the equation for [u]. For a small matrix [a]}

((2

x

2)

or

(3

x

3)

), the eigenvalues can be determined directly by cal­

culating the determinant and solving for the roots of the characteristic equation. This is shown in Example

5-1

_{where the eigenvalue problem}

approach is used for calculating the principal moments of inertia and the directions of the principal axes of an asymmetric cross-sectional area.

Determining the eigenvalues of larger matrices is more difficult. Various numerical methods for solving eigenvalue problems have been developed. Two of them, the power method and the QR factorization method, are described next.

5.3 THE BASIC POWER METHOD

The power method is an iterative procedure for determining the largest real eigenvalue and the corresponding eigenvector of a matrix. Consider an

(

n x n

)

matrix [a] that has n distinct real eigenvalues 'A 1, 'A2, . • . ,'A n

Example 5-1: Principal moments of inertia.

Determine the principal moments of inertia and the orientation of the principal axes of inertia for the cross-sectional area shown.

The moment of inertia

Ix, IY

, and the product of inertia

Ixy

are:

l r-1-°mm

3mm

Ix

= 10228.5 mm4,

IY

= 1307.34 mm4, and

Ixy

= -2880 mm4

SOLUTION

In matrix form, the two-dimensional moment of inertia matrix is given by:

[Ix -Ixy_j

finer

= -J J

xy y

[

10228.5 2880 2880

(5.6)

The principal moments of inertia and the orientation of the principal

axes of inertia can be calculated by solving the following eigenvalue problem:

[Ilner][u]

= A.

[

u

]

�Om�l

(5.7)

where the eigenvalues A. are the principal moments of inertia and the associated eigenvectors

[u]

are unit vectors in the direction of the principal axes of inertia. The eigenvalues are determined by calculating the determinant in Eq.

(5.5):

(5.8)

10228.5 -A.) 2880

J

=

o

2880 (1307.34 -A.)

(5.9)

In document Interacciones socioestatales y presupuesto público (página 41-46)