# What does it mean to have more than one eigen value?

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)?

• had you already seen explicit examples, say in $\Bbb R^2$? Nov 19 '15 at 2:12

You can even get $n$ distinct eigenvalues for an n by n matrix if you just pick any matrix that is zero off the main diagonal and you choose the diagonal entries to be distinct.