---
tag: 04CW1P44VD
kind: statement
role: theorem
name: Spectral theorem for normal matrices
msc: [15A21, 47B15]
uses: [0XYKE8BE7X, 0D6RMJ10M9]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_matrix
    locator: Consequences
---

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then $A$ is [[normal|0MR0BYF0K2]] if and only if there are a [[unitary matrix|0AR68JTMDT]] $U$ 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.