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 lets 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] = lambda*[X] no zero x
value so that we can satisfy this set of equations
[A]=[X] = lambda*[X].
lambda is the eigenvalue and [X] is the eigenvector
so how do we go about doing this in Euler's
method 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 your gonna do is your 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]=lambda*[X1] so
you have [Y1] which you just found out so
the question is how do I find lambda 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 uni uh one you got to keep the same
one to be one and that s how you find lambda
because you have to multiply by scalar 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 lambda values your getting is converging
how do you check for convergence your going
to check for convergence by simply((lambda_1-lambda_i)/lambda_1)*100
for example this is going to give you your
absolute relative approximate error between
the current approximation and the previous
approximation and what your going to do is
your 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
