Mathematical Tools for Physics

7—Operators and Matrices 199

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
(
0
v 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

(36)

7.12 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 constant).
x 1 =K

(

x 11 +x 59 +x 182

)

Similarly the second site’s rating is determined by what links to it, as

x 2 =K

(

x 137 +x 157983 +x 1 +x 876

)