CHAPTER 10.15: EIGENVALUES AND EIGENVECTORS: How does one find eigenvalues and eigenvectors numerically Example
In this segment, letís take an example of finding eigenvalues and vectors numerically. So letís 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 letís 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, letís 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 letís suppose the first component which Iím going to always keep 1. So what Iím going to do is Iím 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? Iím 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 Iím going to say Iíll have to divide by 2.5, so itíll be 2.5 and Iíll get 1 and Iíll 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 donít 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 Iím 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 Ďtil you find out if it is less than your pre-specified tolerance.
So I have a few numbers here so letís look at what we get as we keep on going through the process here so letís suppose I have a table here which Iím 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 thatís 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 letís suppose if I do some more equations letís suppose Iím at the fifth equation so Iím 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 letís suppose 2% error I wouldíve 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 thatís 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 itís unity for all the steps thatís 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 thatís the end of this segment.