Normal Matrix — Definition, Formula & Examples

A normal matrix is a square matrix that commutes with its own conjugate transpose, meaning the order in which you multiply them does not matter.

A square matrix $A \in \mathbb{C}^{n \times n}$ is normal if and only if $A^*A = AA^*$ , where $A^* = \overline{A}^T$ denotes the conjugate transpose of $A$ . For real matrices, this simplifies to $A^TA = AA^T$ .

Key Formula

A^*A = AA^*

Where:

$A$ = A square matrix (real or complex)
$A^*$ = The conjugate transpose of A (equal to A^T for real matrices)

How It Works

To check whether a matrix is normal, compute both products

A^*A

and

AA^*

and see if they are equal. The importance of normal matrices comes from the spectral theorem: a matrix is normal if and only if it is unitarily diagonalizable. This means there exists a unitary matrix

U

such that

A = U D U^*

, where

D

is diagonal. Symmetric, skew-symmetric, orthogonal, Hermitian, skew-Hermitian, and unitary matrices are all special cases of normal matrices.

Worked Example

Problem: Determine whether the real matrix A is normal, where A = [[1, -1], [1, 1]].

Step 1: Since A is real, the conjugate transpose is just the transpose. Compute A^T.

A^T = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}

Step 2: Compute A^T A.

A^T A = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}

Step 3: Compute A A^T.

A A^T = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}

Step 4: Compare the two products. Since A^T A = A A^T, the matrix is normal.

Answer: A is a normal matrix because A^T A = A A^T = 2I.

Why It Matters

Normal matrices are central to the spectral theorem, which guarantees they can be diagonalized by a unitary (or orthogonal) change of basis. This property is used extensively in quantum mechanics, signal processing, and principal component analysis, wherever you need a clean eigenvalue decomposition.

Common Mistakes

Mistake: Assuming every diagonalizable matrix is normal.

Correction: A matrix can be diagonalizable by some invertible matrix without being unitarily diagonalizable. Normality specifically requires A*A = AA*, which is a stronger condition than mere diagonalizability.