Editing Diagonalizable matrix (section)

=== Diagonalizable matrices ===
* [[Involution (mathematics)|Involution]]s are diagonalizable over the reals (and indeed any field of characteristic not 2), with ±1 on the diagonal.
* Finite order [[endomorphism]]s are diagonalizable over <math>\mathbb{C}</math> (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 polynomial|separable]], because the roots of unity are distinct.
* [[Projection (linear algebra)|Projections]] are diagonalizable, with 0s and 1s on the diagonal.
* Real [[symmetric matrices]] are diagonalizable by [[orthogonal matrix|orthogonal matrices]]; i.e., given a real symmetric matrix {{nowrap|<math>A</math>,}} <math>Q^{\mathrm T}AQ</math> is diagonal for some orthogonal matrix {{nowrap|<math>Q</math>.}} More generally, matrices are diagonalizable by [[unitary matrix|unitary matrices]] if and only if they are [[normal matrix|normal]]. In the case of the real symmetric matrix, we see that {{nowrap|<math>A=A^{\mathrm T}</math>,}} so clearly <math>AA^{\mathrm T} = A^{\mathrm T}A</math> holds. Examples of normal matrices are real symmetric (or [[Skew-symmetric matrix|skew-symmetric]]) matrices (e.g. covariance matrices) and [[Hermitian matrix|Hermitian matrices]] (or skew-Hermitian matrices). See [[spectral theorem]]s for generalizations to infinite-dimensional vector spaces.