2018版《应用线性代数与矩阵分析》：实验科学视角下的问题求解工具

Linear

Algebra

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8 1 LINEAR SYSTEMS OF EQUATIONS

The PageRank Tool

Consider the Google problem of displaying the results of a search on a certain

phrase. There could be many thousands of matching web pages. So which

ones should be displayed in the user’s window? Enter PageRank technology

(famously referenced by Kurt Bryan and Tanya Leise in [5] as a “billion dollar

eigenvalue”) which ranks the pages in terms of an “importance” score. This

remarkable technology has found signiﬁcant application in areas such as chem-

istry, biology, bioinformatics, neuroscience, complex systems engineering and

even sports rankings (a comprehensive summary can be found in [13]).

Let’s start small: suppose we have a web of four pages represented as in

Figure 1.4 with pages as vertices and links from one page to another as arrows.

Fig. 1.4: A web with four pages as vertices and links as arrows.

Here is a ﬁrst pass at page ranking (but not the last, we will return to this

signiﬁcant example with reﬁnements several times more in this text). We could

simply count backlinks (incoming links) of each page and rank pages according

to that score, larger being more important than smaller. One problem with

this solution is that it would give equal weight to a link from any page, whether

the linking page were of low or high rank. Another problem is that the rank of

two pages could be artiﬁcially inﬂated by increasing the number of backlinks

and outgoing links between them. So here is a second pass to correct some of

these deﬁciencies: let a page score be the sum of all scores of pages linking

to it. For page i let x

be its score and L

be the set of all indices of pages

linking to it. Then the score for vertex i is given by



∈L

. (1.3)

But that ranking could still give excess inﬂuence to a page simply by its linking

from many other pages. To correct this deﬁciency we make a third pass: for

page j let n

be its total number of outgoing links on that page. Then the

score for vertex i is given by



∈L

. (1.4)

The result is that each page divides its one unit of inﬂuence among all pages to

which it links, so that no page has more inﬂuence to distribute than any other.

1.1 Some Examples 11

Exercise 10. Suppose that in the input–output model of Example 1.5 each

producer charges a unit price for its commodity, say p

and that the

columns of the table represent fraction of each producer commodity needed by

the consumer to produce one unit of its own commodity. Derive equilibrium

equations for these prices.

Exercise 11. Solve the system that results from the second pass of Example 1.6

for page ranking.

Exercise 12. Solve the system that results from the third pass of Example 1.6

for page ranking given that x

is assigned a value of 1.

Exercise 13. Construct a linear system that has x

=1, x

= −1 as a solution

and right-hand side terms b

=1, b

= −2, b

=3.

Exercise 14. Construct a linear system that has both x

=1, x

= −1 and

=2, x

=2as solutions and right-hand side terms b

=3, b

=1, b

=4.

Problem 15. Suppose that we construct a web of pages by removing vertex 4

and its connecting edges from Figure 1.4. Write out the system of equations

that results from the second and third passes of Example 1.6 for page ranking

and solve these systems.

Problem 16. Use ALAMA Calculator or other technology tool to solve the sys-

tems of Examples 1.4 and 1.5. Comment on your solutions. Are they sensible?

Problem 17. A polynomial y = a

+ a

x + a

is required to interpolate a

function f(x) at x =1, 2, 3,wheref(1) = 1, f(2) = 1,andf(3) = 2. Express

these three conditions as a linear system of three equations in the unknowns

. What kind of general system would result from interpolating f (x)

with a polynomial at points x =1, 2,...,n where f (x) is known?

*Problem 18. The topology of a certain network is indicated by the digraph

(directed graph) pictured below, where ﬁve vertices represent locations of

hardware units that receive and transmit data along connection edges to other

units in the direction of the arrows. Suppose the system is in a steady state

and that the data ﬂow along edge j is the nonnegative quantity x

. The single

law that these ﬂows must obey is this: net ﬂow in equals net ﬂow out at each

of the ﬁve vertices (like Kirchhoﬀ’s ﬁrst law in electrical circuits). Write out

a system of linear equations satisﬁed by variables x

12 1 LINEAR SYSTEMS OF EQUATIONS

Problem 19. Use ALAMA Calculator or other technology tool to solve the

system of Example 1.3 with conductivity K =1and internal heat source

f(x)=x and graph the approximate solution by connecting the points (x

)

as in Figure 1.3.

1.2 Notation and a Review of Numbers

The Language of Sets

The language of sets pervades all of mathematics. It provides a convenient

shorthand for expressing mathematical statements. Loosely speaking, a set

can be deﬁned as a collection of objects, called the members of the set. This

deﬁnition will suﬃce for us. We use some shorthand to indicate certain rela-

tionships between sets and elements. Usually, sets will be designated by upper-

case letters such as A, B, etc., and elements will be designated by lowercase

letters such as a, b,etc.Asusual,setA is a subset of set B if every element of

A is an element of B,andaproper subset if it is a subset but not equal to B.

Two sets A and B are said to be equal if they have exactly the same elements.

Set Symbols

Some shorthand:

∅ denotes the empty set, i.e., the set with no members.

a ∈ A means “a is a member of the set A.”

A = B means “the set A is equal to the set B.”

A ⊆ B means “A is a subset of B.”

A ⊂ B means “A is a proper subset of B.”

Therearetwowaysinwhichwemaydeﬁneaset:wemaylist its elements,

such as in the deﬁnition A = {0, 1, 2, 3}, or specify them by rule such as in

the deﬁnition A = {x | x is an integer and 0 ≤ x ≤ 3}. (Read this as “A is

the set of x such that x is an integer and 0 ≤ x ≤ 3.”) With this notation we

can give formal deﬁnitions of set intersections and unions:

Deﬁnition 1.3. Set Union, Intersection, Diﬀerence Let A and B be sets.

Then the intersection of A and B is deﬁned to be the set A ∩ B =

{x | x ∈ A and x ∈ B}.Theunion of A and B is the set A ∪ B =

{x | x ∈ A or x ∈ B} (inclusive or, which means that x ∈ A or x ∈ B or

both). The diﬀerence of A and B is the set A − B = {x | x ∈ A and x ∈ B}.

Example 1.7. Let A = {0, 1, 3} and B = {0, 1, 2, 4}.Then

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