Hello students I have taken my first video on
rank of matrix and my second video
was on consistent and inconsistent. Today
I am going to tell you that what is eigen value and eigen
vectors. So for today
my thought is that keep working hard and be ready.
You all might make certain mistakes but these problems
as you work hard
you all will make less mistakes. With this thought today
I am ready with another video for you all. So
what is eigen value and eigen vectors of the
matrices? Students face a lot of problem in this topic and in engineering
bsc students and competitive students in all this is asked.
So today I will teach you what are eigen values and eigen vectors but this whole corrected topic
is from rank and even has come connection with consistent and in consistent.
So first I will tell you the definition
that how do we find the eigen value and eigen vectors
and what are they? So first the definition
is what we have to see and try to understand that how we find the eigen values and eigen vectors.
and what is the definition behind this? So
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
So the definition of eigen values that comes, what it is? it is the characteristic equation
and when we put there value as 0 the roots that we get
these roots are called as the eigen value of the
matrix. After this we have the concept of the eigen vectors.
So what is eigen vector?
For the eigen values that we get, for that values
we have some corresponding eigen vecto.
 
 
 
 
So we call this solution as corresponding eigen
vectors. This is the definition and the definition will
be much clear when I will do the questions that what it is trying to say. So we will take one question here.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Students now I have told you how we find the eigen value and eigen vectors but I see that
the questions in GATE that are based on engineering mathematics that
you are given four eigen values
options and you have to find its eigen vector or you are given four options
and you have to find that this eigen vector is corresponding to this eigen value so I am telling you a trick.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
So today I have told you two questions related to eigen value and eigen vectors.
I have even told you the definition for eigen value and eigen vectors and what are they?
So first here we have read the concept that
what is characteristic matrix and after that we have studied about characteristic equation and
after characteristic equation the roots that we get
are the eigen values and if we keep the  eigen values in the characteristic matrix so the solution that we get
the non zero solution solution we call it eigen
vector. Here I have told you two examples
and I want to tell you again that if there are three matrices and all the three eigen values are non zero
then the rank is three and if the rank is three
and if we put one eigen value into it the value decreases by 1
so from that we get the infinite solutions.
These infinite solutions are the eigen vectors.
Here we can even check if our eigen values are correct or not.
If you find the wrong eigen values then your eigen vectors will be 100% wrong.
So students keep checking that the question that you are doing as this question is quite long and in long questions chances of mistakes
are high so in between
we have to apply the check points that are we doing correct or not.
So as soon as you find the eigen values do check that the eigen values
are not wroung.
So when you check the eigen values we add all the three
so on adding, to its diagonal sum
it should be equal and it is the coefficient of the lamda square.
There is one more concept that if we apply the three eigen values
so it is equal to its determinant or
the constant of the characteristic equation
it is equal to that. So this is the concept.
I have even explained you the short trick for the eigen vectors that
if we have to check the eigen vector that this eigen vector
is corresponding to which eigen value as there are three eigen values
so there will be three eigen vectors. So
if in GATE a question is asked or a question is asked in any exam
and they ask that this eigen vector is
corresponding to which eigen value? So what you have to do is
that in that
you will check with the help of trick that was told that this eigen value
that this eigen vector is corresponding to which eigen value. Thank you students. Thank you for watching.
If you have any doubt related to this subject
eigen values and eigen vectors. Before this I have made a video on rank and before that on consistent
and inconsistent after that I have made the video.
So in that you have any doubt then in my comment box
do write your comment. Do subscribe to our channel
and do share. My main objective is for engineering students and for there benefit that
without any problem they understand the concept.
The main problem that comes is that students in the classes
students do study but the concept they are trying to understand they are able to understand in that way. All
these subjects even has applications in your engineering also.
Even in technical subjects also they have the application in the sense
you can see in the electrical or in electronics
matrices are used a lot. Even in computer science also it is used a lot.
So I feel that you won't get any problem in understanding.
But then also if you feel that in my teaching
I should explain like this or if there is any problem with the board or
if there is any problem in my writing then do comment and tell me.
Thank you. Thank you for watching. Thanks!
