CHAPTER 10.15: EIGENVALUES AND
EIGENVECTORS: How does one find eigenvalues and eigenvectors numerically
Example In
this segment, let’s take an example of finding eigenvalues and vectors
numerically. So let’s suppose the problem stating use power method to find
largest eigenvalue by magnitude and corresponding 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’s 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’s 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’s suppose the first component which I’m
going to always keep 1. So what I’m going to do is I’m going to do [A] times
[Xnot] 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 pre-specified
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 λ 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
λ 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’s 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 AX is equal to lambda times X will be your eigenvector. And
that’s the end of this segment. |