> Fixed points, diagonalizations and eigen… are all the same thing. This is true in a moral sense, but not literally. Fixed points are eigenvectors for a part

> Fixed points, diagonalizations and eigen… are all the same thing. This is true in a moral sense, but not literally. Fixed points are eigenvectors for a particular eigenvalue \lambda = 1 (together with the 0 vector, which is always fixed but not an eigenvector, by fiat). (Over a field,) eigenvectors corresponding to non-0 eigenvalues are not fixed points, although they lie on fixed lines; and they can be made fixed points by re-scaling the original matrix, if that is convenient. I don't know an obvious way to think of eigenvectors with eigenvalue \lambda = 0 as fixed vectors. A matrix that is diagonalizable has sufficiently many eigenvectors, but a matrix may have eigenvectors even if it is not diagonalizable.