Lie algebra. Examples Diagonalizable matrices Involutions are diagonalizable over the reals (and indeed any field of characteristic not 2), with ±1 on the diagonal. Finite order endomorphisms are diagonalizable over (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with roots of unity on the diagonal. This follows since the minimal polynomial is separable, because the roots of unity are distinct. Projections are diagonalizable, with 0s and 1s on the diagonal. Real symmetric matrices are diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix is diagonal for some orthogonal matrix More generally, matrices are diagonalizable by unitary matrices if and only if they are normal. In the case of the real symmetric matrix, we see that so clearly holds. Examples of normal matrices are real symmetric (or skew-symmetric) matrices (e.g. covariance matrices) and Hermitian matrices (or skew-Hermitian matrices). See spectral theorems for generalizations to infinite-dimensional vector spaces. Matrices that are not diagonalizable In general, a rotation matrix is not diagonalizable over the reals, but all rotation matrices are diagonalizable over the complex field. Even if a matrix is not diagonalizable, it is always possible to "do the best one can", and find a matrix with the same properties consisting of eigenvalues on the leading diagonal, and either ones or zeroes on the superdiagonal – known as Jordan normal form. Some matrices are not diagonalizable over any field, most notably nonzero nilpotent matrices. This happens more generally if the algebraic and geometric multiplicities of an eigenvalue do not coincide. For instance, consider This matrix is not diagonalizable: there is no matrix such that is a diagonal matrix. Indeed, has one eigenvalue (namely zero) and this eigenvalue has algebraic multiplicity 2 and geometric multiplicity 1. Some real matrices are not diagonalizable over the reals. Consider for instance the matrix The matrix does not have any real eigenvalues, so there is no real matrix such that is a diagonal matrix. However, we can diagonalize if we allow complex numbers. Indeed, if we take then is diagonal. It is easy to find that is the rotation matrix which rotates counterclockwise by angle Note that the above examples show that the sum of diagonalizable matrices need not be diagonalizable. How to diagonalize a matrix Diagonalizing a matrix is the same process as finding its eigenvalues and eigenvectors, in the case that the eigenvectors form