TITLE: A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
QUESTION [77 upvotes]: A number of areas I'm studying in my degree (not a maths degree) involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the explanations given in textbooks and lectures. Does anyone know of a good, fairly simple but mathematical explanation of eigenvectors and eigenvalues on the internet? If not, could someone provide one here?
As well as some of the mathematical explanations, I'm also very interested in 'big picture' answers as to why on earth I should care about eigenvectors/eigenvalues, and what they actually 'mean'.

REPLY [4 votes]: This video has a good simple explanation that also includes math.
[2016, 3Blue1Brown, "Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14"]
My very loose interpretation of the 'meaning' from that video is as follows:
Eigenvectors can be visualized as a line that's an axis of rotation, where no matter how much things change around it, the angle / direction of this line doesn't change.
I visualize it first like a fan, where regardless of how much the blades spin, its centerpoint ("Eigenvector") is still pointed in the same direction.
In a matrix that involves a transformation,
it can also be thought of as a kind of rotation, but one where the rotation can change the lengths of vectors within a zone of rotation.
After a transformation, there can still be one or more such lengths that don't change in direction (Eigenvectors), but will change in size.
The change in size would be the "Eigenvalue", like an Eigenvalue of 2 indicates that the length of the Eigenvector had doubled after transformation and 1/2 would mean it had reduced by half.
The picture below, based on that video link, would be an illustration of the above explanation.