I am trying to understand eigenvectors. An Eigenvector is nothing more than a vector that points to some place. This pointing vector will then be invariant under linear transformations.
Now my questions:
- Ok so this vector is invariant. So what? (in my case for attitude determination algorithm I even less understand what this could give me as useful information)
- how does a simple $4\times 4$ matrix actually represent a transformation?
No, an eigenvector is a vector that points to the same place after being transformed by the matrix.
As far as usefulness of eigenvectors go, it is very hard to find a mathematical concept that has more real life applications then the concept of eigenvectors: for example, the PageRank algorithm, which finds the dominant eigenvector of a matrix, is one of the most important parts in Google's algorithms for ranking webpages. Other examples include:
And many more. Each of these fields has it's own problems, but all of these problems can be translated into finding an eigenvector for a specific matrix. The "meaning" behind these vectors is then bestowed upon the solutions not by algebra ("pointing in the same direction"), but by the field that first had the problem!