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.