CHAPTER 10.03: EIGENVALUES AND
EIGENVECTORS: Definition of eigenvalues and eigenvectors In
this segment, we are going to talk about what the definition of eigenvalues
and eigenvectors is. So if we have [A] is a square matrix, n by n matrix,
then we look at our vector [X] and we say it's not equal to zero is an
eigenvector of [A] if [A][X]= λ[X] so if you have a square matrix and
you find out that hey there is a vector a column vector which is not a zero
vector and if that satisfies this particular condition here that [A][X]=
λ[X] then [X] is called the eigenvector of [A] and λ is the eigenvalues
and of course λ is a scalar, so it's just a number. So that's how we
define eigenvalues and eigenvectors so all you ever do is you have to find a
vector a column vector, which is non-zero, so that when you multiply it to
the [A] matrix that it turns out to be some number times the eigenvector
itself or this vector, column vector [X] itself and whatever that scaler is
by which you are multiplying it so that this equality is held good is called
an eigenvalue so λ is the eigenvalue so you can get different
eigenvalues for an n by n matrix so if you have an n by n matrix you'll get n
eigenvalues and corresponding to each eigenvalue you have an eigenvector so
an n by n matrix has n eigenvalues, which we may call λ1, λ2, all
the way up to λn they are not necessarily
going to be unique but we will have n different eigenvalues and corresponding
to each eigenvalue you will have an eigenvector so those are things which we
have to think about when we talk about what it means for a particular square
matrix to have eigenvalues and eigenvectors. And that's the end of this
segment. |