Spectral theorem for normal matrices

Statement

Let $A$ be an $n \times n$ complex matrix. Then $A$ is normal if and only if there are a unitary matrix $U$ and a diagonal matrix $D$ with $$A = U D U^*.$$

Remarks

The columns of $U$ are then an orthonormal basis of $\C^n$ of eigenvectors of $A$. If $A$ is Hermitian, then $D$ is real; if $A$ is unitary, then the diagonal entries of $D$ have absolute value $1$. Over $\R$, a normal matrix need not be diagonalisable: a rotation of the plane is normal.

Explanations

L1 human

The normal matrices are exactly the complex matrices that are diagonal in some orthonormal basis. This generalises the spectral theorem from Hermitian matrices to all normal matrices. The eigenvalues can now be any complex numbers.