(Preliminary work. Use this in the next questions!) If A is a n imes n matrix, then the
equation Ax=0_(n imes 1) always has the solution x=0_(n imes 1), called the trivial solution.
However, there may be other solutions: for instance, if
A=([1,-2],[-2,4]),
then x=([2],[1]) is a (non-trivial) solution to Ax=0_(n imes 1). Now, show that if A is
invertible, then the equation Ax=0_(n imes 1) only has the trivial solution. Note that the
converse is also true: if A is not invertible, then the system has (infinitely many)
non-trivial solutions.
In the rest of the exercise, we consider a very simple example with three web pages
P_(1),P_(2),P_(3), where
P_(1) references P_(2) only;
P_(2) references P_(1) and P_(3);
P_(3) references P_(1) and P_(2).
The popularity of page P_(1),P_(2),P_(3) is denoted by x,y,z. As a first model, we assume
that the popularity of each page is the sum of the popularity of each page that it is
referenced by. This gives rise to the equations
x=y+z
y=x+z
z=y.
Find the solution(s) to this system in any way that you want. There are several
ways to do this, but you can do so with barely any calculations.
You should see that the previous model is not very satisfying! Instead, assume that
the popularity of each page is a fraction lambda of the sum of the popularity of each page
that it is referenced by. This gives rise to the equations
x=lambda (y+z)
y=lambda (x+z)
z=lambda y.
Find the solution to this system in any way that you want for lambda =(1)/(10) (each page
gets 10% of the popularity of the pages it is referenced by). Comment.
Under what condition on lambda does the system of Q3 have a nontrivial solution? You
should give an equation involving lambda , but there is no need to solve it.
You could solve the previous equation for lambda and find solutions lambda =-1,lambda ~~-1.618,
and lambda ~~0.618. The negative lambda are not relevant, since they would give some negative
values for x,y, or z, which does not make sense. Therefore, you decide to solve
the system with lambda =0.618. You plug this system of equations into your favorite
calculator or computing software. It tells you (try it!) that the system only has the
trivial solution x=y=z=0. Why?
1. (Preliminary work. Use this in the next questions!) If A is a n n matrix, then the equation AX = 0nx1 always has the solution X = On1, called the trivial solution. However, there may be other solutions: for instance, if
then X =
is a (non-trivial) solution to AX = 0nx1. Nowshow that if A is
invertible, then the equation AX = On1 only has the trivial solution. Note that the converse is also true: if A is not invertible, then the system has (infinitely many) non-trivial solutions.
2. In the rest of the exercise, we consider a very simple example with three web pages P1, P2, P3, where
P references P2 only; P references Pand P3 P3 references Pand P2
The popularity of page P1, P2, P3 is denoted by , y, z. As a first model, we assume that the popularity of each page is the sum of the popularity of each page that it is referenced by. This gives rise to the equations
y+z +2 y.
Find the solution(s) to this system in any way that you want. There are several ways to do this, but you can do so with barely any calculations.
3. You should see that the previous model is not very satisfying! Instead, assume that the popularity of each page is fraction A of the sum of the popularity of each page that it is referenced by. This gives rise to the equations
X(y+ 2) 3= X(x + 2) Xy
3
Find the solution to this system in any way that you want for = 1/10 (each page gets 10% of the popularity of the pages it is referenced by). Comment.
4. Under what condition on A does the system of Q3 have a nontrivial solution? You should give an equation involving A, but there is no need to solve it.
5. You could solve the previous equation for A and find solutions A= -1, A ~ -1.618, and ~ 0.618. The negative A are not relevant, since they would give some negative values for , y, or z, which does not make sense. Therefore, you decide to solve the system with A = 0.618. You plug this system of equations into your favorite calculator or computing software. It tells you (try it!) that the system only has the trivial solution = y = z = 0. Why?
