I am struggling to understand, conceptually, what it means for a matrix to have more than one eigen value. I know that eigen values are scalars that correspond to eigen vectors, and that eigen vectors are those that, when put through a certain linear transformation, come out as scalar multiples of themselves.
What I'm actually confounded by is how any matrix can have a determined eigen value at all. That is to say, how can you know which scalar multiples of any vector you're going to get when you put it through your linear transformation? Is it true that one matrix is going to scale any kind of vector in a certain kind(s) of way(s)?
I'd appreciate any help you may be able to provide.