CHAPTER 10.14: EIGENVALUES AND EIGENVECTORS: How does one find eigenvalues and eigenvectors numerically?

In this segment we will talk about how to find eigenvalues and eigenvectors numerically this of course um all of this is an introduction to matrix algebra course most of the times when you find eigenvalues and eigenvectors numerically so I just want to give you a flavor of how we do we find it numerically and one of the methods which is used to find eigenvalues and eigenvectors is called the Power Method now the power method only finds the largest eigenvalue in magnitude so if you would know the eigenvalues of a square matrix and take the absolute value of all of them it will find the one for which that absolute value is largest it will find only that eigenvalue so you are going to find the largest eigenvalue magnitude and also it does not work it does not work for repeated eigenvalues so if the largest eigenvalue is repeated then it this power method will not work.

So let’s go and see that how this is going to work for us the problem which we have is that we want to find the eigenvalues and eigenvectors by doing this [A]=[X] = λ[X] no zero x value so that we can satisfy this set of equations [A]=[X] = λ[X] λ is the eigenvalue and [X] is the eigenvector so how do we go about doing this in Euler’s vector is that we assume a guess [X] so that’s a guess for the eigenvector choose one component to be unity so what that means is that the guess which you have chosen of course it has to be a nonzero vector because that’s how the eigenvector is defined eigenvector has to be a nonzero vector but one of the components of this vector [X] has to be unity has to be 1 so and then you have to keep it to be one throughout the whole process so you choose that to be one so once you have chosen that to be one what you’re going to do is you’re going to find [Y1] another vector [Y1] which will be found by simply taking the [X] vector which you just assumed which you use as a guess as your first estimate for an eigenvector [Y1]=[A][X] so how do you find X1 now you find X1 by saying hey [Y1]=λ[X1] so you have [Y1] which you just found out so the question is how do I find λ equal to some other vector X1 and the way you do it is you say by keeping same component to be unity so whichever component you chose to be one you got to keep the same one to be one and that’s how you find λ because you have to multiply by scaler so that the same component becomes unity in this one so what that means is that you have found the next guess for your eigenvector so now it becomes a repetition part so repeat steps two and three until convergence so you will repeat steps two and three until you find out if the λ values your getting is converging how do you check for convergence your going to check for convergence by simply( λ1-λi÷λ1)*100 for example this is going to give you your absolute approximate error between the current approximation and the previous approximation and what you’re going to do is you’re going to check whether it is less than pre-specified tolerance so you might have specified tolerance of .5%, .25% based on that you will be able to use that as your stopping criteria so we will look at this whole algorithm through an example and that’s how you find the eigenvalue of a matrix and this is the end of this segment