In this segment, let us take an example of finding eigenvalues and vectors numerically.
So let us suppose the problem stating use power method to find largest eigenvalue by magnitude and correspondin
vector, corresponding eigenvector of this matrix. So somebody is giving you the [A]
matrix and is asking you to find that. So you have 1.5, 0, 1, -0.5, 0.5, -0.5, -0.5,
0 and 0. So let us see how that works here. So what we are going to do is, we are going
to take initial guess so this is the problem statement. So let your initial guess, let
us suppose, be equal to 1, 1, 1 and what we have to do is, we have to assume one of them,
of course when we make the initial guess of a non-zero starting eigenvector, one of the
components has to be one. Although we chose 1, 1, 1, as all of the components let suppose
the first component which I am going to always keep 1. So what I m going to do is I am going
to do [A] times [Xo] my initial guess so it will be 1.5, 0, 1, -0.5, 0.5, -0.5, -0.5,
0, 0 and I multiply by 1, 1, 1. What am I going to get? I m going to get 2.5, -0.5, and -0.5. But what I want to
do is, I want to keep the first component of this eigenvector to be always one so I
m going to say I ll have to divide by 2.5, so it ll be 2.5 and I ll get 1 and I ll get
-0.2 and -0.2 so that is my first initial guess of the first guess which I get for the
eigenvalue. And this is the first guess I get for the eigenvector of course. But I don
t have anything to compare it with as far as the eigenvalue is concerned so I am going
to go for one more equation I suppose. So now my [X] of 1 is equal to whatever I get
there 1, -0.2, and, -0.2. I will do [A] times [X] of 1 and I will multiply the [A] vector
which is this and I m going to multiply it by the [X] vector which I found 1, -0.2, -0.2
and when I do the matrix multiplication, this is what it turns out to be it turns out to
be: 1.3, -0.5, and -0.5 and since I said that the first component of this eigenvector is
always going to be 1 this is because 1.3 times 1, -0.3846, -0.3846 so this is my this is
my second estimate of the eigenvalue so now what I can do is I can compare this number
and this number right here to see that how close I am coming to the eigenvalue. So we
have the first estimate as 2.5 as our first estimate and 1.3 as our next estimate so what
is the relative approximate here it will be the current approximations 1.3 minus the previous
approximation 2.5 over the 1.3 and when you multiply it by 100 that number turns out to
be 92.307% so that seems to be a large error so what you are going to do is you are going
to keep on doing this process til you find out if it is less than your prespecified tolerance.
So I have a few numbers here so let s look at what we get as we keep on going through
the process here so let s suppose I have a table here which I m going to make for what
step I am at what is the value of lambda I am calculating, what is the corresponding
value of the eigenvector I am calculating, and what is my absolute relative approximate
error in percentage in terms of the eigenvalues and so we already saw that hey when I do my
first equation my eigenvalue turns out to be 2.5 and the corresponding eigenvector turns
out to be 1, -0.2, and -0.2 and I cannot calculate my relative approximate error here so that
s my first one if I look at my second one which I just calculated as 1.3 and the corresponding
eigenvector turns out to be 1, -0.3846, -0.3846 and the relative approximate turned out to
be 92.307% so I can continue doing this process and I want to find out let s suppose if I
do some more equations let s suppose I m at the fifth equation so I m skipping showing
you the values for 3 and 4 but at the end of the fifth equation I get 1.02459 as my
eigenvalue the eigenvector corresponding to that I get is 1, -0.4880, -0.4880 I get here
and the relative approximate which I get is 1.2441% so if somebody had asked me hey I
want you to calculate your eigenvalue within let s suppose 2% error I would've stopped
here saying that hey I get 1.2441% as my relative approximate error and this is my estimate
of eigenvalue and this is my estimate of the eigenvector now if you were to have kept on
going through this process you will found out that from the exact value turns out to
be the lambda eigenvalue the largest eigenvalue is 1 and the eigenvector which will have the
first value as 1 would have been 1, -0.5, -0.5 so if you keep on going through this
process you will get closer and closer to the exact value of 1 and the eigenvector of
1, -0.5, and -0.5 and that s how the power method works so keep in mind that the power
method only finds the largest eigenvalue in magnitude and the way to continue the iterations
is to choose one of the components to be unity and keep on and insist that it is unity for
all the steps that s how we are going to get the conversions to that particular eigenvalue
and whatever is the eigenvector the vector corresponding to the satisfaction of [A][X]=lambda*[X]
will be your eigenvector. And that's the end of this segment.