One way to think of eigenvectors: Your linear map A moves things around, and you aim to characterise the linear map. So, look for lines (through the origin) t

One way to think of eigenvectors: Your linear map A moves things around, and you aim to characterise the linear map. So, look for lines (through the origin) that are not moved. Those are given by eigenvectors. A point on that line might be moved closer to or further away from the origin (depends on eigenvalue < or > 1), or even flipped to the other side (if eigenvalue < 0), but the line as a whole is mapped to itself.