L48: diagonalization

Definition of diagonalizable linear map and of what it means for a matrix to be diagonalizable over R or C. A sufficient condition for diagonalizability of a linear map on a n-dimensional vector space is that it has n distinct eigenvalues, but this is not necessary (e.g. the identity is always diagonalizable even though it only ever has one eigenvalue). Matrix of a linear map with respect to a basis of eigenvectors. If P is a matrix whose columns are eigenvectors of the matrix A then P^{-1}AP is diagonal. Examples of diagonalizable and non-diagonalizable linear maps. Finding powers of diagonal matrices is easy, examples of why you might want to find them.