What is an eigenspace? No video or anything out there really explains what an eigenspace is? From what i have understood it is just a direction. But why do we need it?.

The following questions have been bugging me for quite a while, and I can't find a real straight forward answer to them. Hopefully one of you can help me.

- What is an eigenspace?
- Why are the eigenvectors calculated in a diagonal?
- What is the practical use of the eigenspace? Like what does it do or what is it used for? other than calculating the diagonal of a matrix.
- why is it important o calculate the diagonal of a matrix?

I want to know mainly because I just passed a linear algebra course and I have no idea what eigenspace is. Which is embarrassing for me and for my professor because he didn't explain what they were, he just basically said: "This is how you calculate it and if you want to know more then read about it in the book ".