340 CHAP. 8 Linear Algebra: Matrix Eigenvalue Problems
8.2 Some Apptications of Eigenvalue Problems
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kind as our last example.
EXAMPLE I Stretching of an Elastic Membrane
An elastic membrane in the xp2-plane with boundary círcle x12 l x22 : 1 (Fig. 158) is stretched so that a
point P: (xt, xz) goes over into the point Q: (yy yz) given by
(|) ,: [''-]
: o*:
Ljz)
5x1 -| 3x2: i"r
(2) or
3x1 -| 5x2: Árz
The characteristic equation is
-3x1 * 3x2- 0,
3x1 - 3x2: 0.
For i2 : 2, our system (2) becomes
3x1 l 3x2: 0,
3x1 -| 3x2: 0.
ln components,
(5-.\)x1 + 3xz :0
3h + (5 - l)xr: g.
[:;][";]
}r:5xr+3x2
];z: 3xt + 5x2.
Find the principal directions, that is, the directions of the position vector x of P for which the direction of the
position vector y of Q is the same or exactly opposite. What shape does the boundary circle take under this
deformation?
SOlUtiOn. We are looking for vectors x such that y : .trx. Since y : Ax, this gives Ax : ),x, the equation
of an eigenvalue problem. In components, Ax : ,i'x is
(3)
l5-^ 3 l
l, 5_^l
:,r-N2-9:0.
Its solutions are i1 : 8 and hz : 2. These are the eigenvalues of our problem. For i - il : 8, our system
(2) becomes
Solution x2: xy x1 arbitrary,
for instance. x1 : x2: 7.
Solution x2: -x1, .h arbitrary,
for instance. xr : | . xz : -|.
We thus obtain as eigenvectors of A, for instance, [1 1]T corresponding to.[1 and [1 -1]T corresponding to
i2 (or a nonzero scalar multiple of the se). These vectors make 45' and 135" angles with the positive xl-direction.
They give the principal directions, the answer to our problem. The eigenvalues show that in the principal
directions the membrane is stretched by factors 8 and 2, respectively; see Fig. 158.
Accordingly, if we choose the principal directions as directions of a new Cartesian ulu2-coordinate system,
say, with the positive ,l-semi-axis in the first quadrant and the positive u2-semi-axis in the second quadrant of
the xlr2-system, and if we set ul: f cos @, u2: r sin @, then a boundary point of the unstretched circular
membrane has coordinates cos Q, sín Q. Hence, after the stretch we have
zr:8cos@, z2:2sinQ.
Since cos2 @ + sin2 ó : 1, this shows that the deformed boundary is an ellipse (Fig. 158)
(4)
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^rT
tJ-
2
Z,)
_-l
a2L
tr
SEC. 8.2 Some Applications of Eigenvalue Problems
Fig.l58. Undeformed and deformed membrane in Example 1
EXAMPLE 2 Eigenvalue Problems Arising from Markov Processes
Markov processes as considered in Example 13 of Sec. 7 .2 lead to eigenvalue problems if we ask for the limit
state of the process in which the state vector x is reproduced under the multiplication by the stochastic matrix
A governing the process, that is, Ax : x. Hence A should have the eigenvalue 1, and x should be a corresponding
eigenvector. This is of practical interest because it shows the long-term tendency of the development modeled
by the process.
In that example,
For the transpose,
Hence AT has the eigenvalue 1, and the same is true for A by Theorem 3 in Sec. 8.1, An eigenvector x of A
for ), : 1 is obtained from
row-reduced to
I
li,.EX,fi,ff:P:[ ,3
Takingí3:1,wegetxz:6from-12/30*r3l5:0andthenx1 :2from-3x1/10*x2l10:0.This
gives x : [2 6 1]T. It means that in the long nrn, the ratio Commercial:Industrial:Residential will approach
2:6:I, provided that the probabilities given by A remain (about) the same. (We switched to ordinary fractions
to avoid rounding errors.) l
Eigenvalue Problems Arising from Population Models. Leslie Model
The Leslie model describes age-specified population growth, as follows. Let the oldest age attained by the
females in some animal population be 9 years. Divide the population into three age classes of 3 years each. Let
íhe "Leslie matrix" be
where /17. is the average number of daughters born to a single female during the time she is in age class k, and
l1,i_t(j :2,3) is the fraction of females in age class j - 1 that will survive and pass into class j. (a) What is
the number of females in each class after 3, 6, 9 years if each class initially consists of 400 females? (b) For
what initial distribution will the number of females in each class change by the same proportion? What is this
rate of change?
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