Database Reference
In-Depth Information
We must solve
When we multiply the matrix and vector we get two equations
3
x
+ 2
y
= 7
x
2
x
+ 6
y
= 7
y
Notice that both of these equations really say the same thing:
y
= 2
x
. Thus, a possible ei-
genvector is
But that vector is not a unit vector, since the sum of the squares of its components is 5, not
1. Thus to get the unit vector in the same direction, we divide each component by That
is, the principal eigenvector is
and its eigenvalue is 7. Note that this was the eigenpair we explored in
Example 11.1
.
For the second eigenpair, we repeat the above with eigenvalue 2 in place of 7. The equa-
tion involving the components of
e
is
x
= −2
y
, and the second eigenvector is
Its corresponding eigenvalue is 2, of course.
□
11.1.3
Finding Eigenpairs by Power Iteration
We now examine the generalization of the process we used in
Section 5.1
to find the prin-
cipal eigenvector, which in that section was the PageRank vector - all we needed from
among the various eigenvectors of the stochastic matrix of the Web. We start by computing
the principal eigenvector by a slight generalization of the approach used in
Section 5.1
. We
then modify the matrix to, in effect, remove the principal eigenvector. The result is a new
matrix whose principal eigenvector is the second eigenvector (eigenvector with the second-
largest eigenvalue) of the original matrix. The process proceeds in that manner, removing
each eigenvector as we find it, and then using power iteration to find the principal eigen-
vector of the matrix that remains.
