TITLE: Diagonalisability of 2×2 matrices with repeated eigenvalues
QUESTION [16 upvotes]: In general for all real (and complex ) 2 by 2 matrices, is it true that if there is a repeated eigenvalue (so all eigenvalues are the same), then we conclude that that matrice is non diagonalisable?  
In general, for a n by n matrix, if ALL eigenvalues are repeated, then does this mean that matrix is non diagonalisable? Does the proof have to do with that if there was a diagonal matrix, then itd be a multiple of the n by n identity matrix, so there is no invertible matrix that satisfies the definition of similarity (diagonalisability)?

REPLY [11 votes]: To be more precise than the previous answers: Repeated eigenvalues are necessary but not sufficient for a matrix to not be diagonalizable.
For example you can never have a $2\times 2$ matrix with two different eigenvalues which is not diagonalizable.
In other words: as soon as all eigenvalues are distinct then we can be sure to be able to diagonalize it.