7—Operators and Matrices 169with familiar roots
α=
(
−b±
√
b^2 − 4 km
)
/ 2 m
If the two roots are equal youmaynot have distinct eigenvectors, and in this case youdonot. No
matter, you can solve any such problem for the case thatb^2 − 4 km 6 = 0and then take the limit as this
approaches zero.
The eigenvectors come from the either one of the two equations represented by Eq. (7.60). Pick
the simpler one,αA=B. The column matrix
(
A
B
)
is thenA
(
1
α
)
.
(
x
vx
)
(t) =A+
(
1
α+
)
eα+t+A−
(
1
α−
)
eα−t
Pick the initial conditions thatx(0) = 0andvx(0) =v 0. You must choosesomeinitial conditions in
order to apply this technique. In matrix terminology this is
(
0v 0
)
=A+
(
1
α+
)
+A−
(
1
α−
)
These are two equations for the two unknowns
A++A−= 0, α+A++α−A−=v 0 , so A+=
v 0
α+−α−
, A−=−A+
(
x
vx
)
(t) =
v 0
α+−α−
[(
1
α+
)
eα+t−
(
1
α−
)
eα−t
]
If you now take the limit asb^2 → 4 km, or equivalently asα−→α+, this expression is just the definition
of a derivative.
(
x
vx
)
(t)−→v 0
d
dα
(
1
α
)
eαt=v 0
(
teαt
(1 +αt)eαt
)
α=−
b
2 m
(7.61)
7.13 Eigenvalues and Google
The motivating idea behind the search engine Google is that you want the first items returned by a
search to be the most important items. How do you do this? How do you program a computer to
decide which web sites are the most important?
A simple idea is to count the number of sites that contain a link to a given site, and the site that
is linked to the most is then the most important site. This has the drawback that all links are treated
as equal. If your site is referenced from the home page of Al Einstein, it counts no more than if it’s
referenced by Joe Blow. This shouldn’t be.
A better idea is to assign each web page a numerical importance rating. If your site, #1, is
linked from sites #11, #59, and #182, then your rating,x 1 , is determined by adding those ratings
(and multiplying by a suitable scaling constant).
x 1 =C
(
x 11 +x 59 +x 182
)
Similarly the second site’s rating is determined by what links to it, as