Hermitian Matrix — Definition, Formula & Examples

A Hermitian matrix is a square matrix with complex entries that equals its own conjugate transpose. This means if you take the transpose and then complex-conjugate every entry, you get the original matrix back.

A square matrix $A \in \mathbb{C}^{n \times n}$ is Hermitian if $A = A^*$ , where $A^* = \overline{A}^T$ denotes the conjugate transpose. Equivalently, for all indices $i$ and $j$ , the entry $a_{ij} = \overline{a_{ji}}$ . This forces the diagonal entries to be real, since $a_{ii} = \overline{a_{ii}}$ .

Key Formula

A = A^* \quad \text{where} \quad A^* = \overline{A}^{\,T}

Where:

$A$ = A square matrix with complex entries
$A^*$ = The conjugate transpose of A (also written A^\dagger)
$\overline{A}$ = The matrix obtained by complex-conjugating every entry of A

How It Works

To check whether a matrix is Hermitian, first transpose it (swap rows and columns), then take the complex conjugate of every entry. If the result is identical to the original matrix, it is Hermitian. A key consequence is that all eigenvalues of a Hermitian matrix are real numbers, even though the matrix entries may be complex. Additionally, eigenvectors corresponding to distinct eigenvalues are orthogonal. For a real-valued matrix, the conjugate transpose is just the ordinary transpose, so a real Hermitian matrix is simply a symmetric matrix.

Worked Example

Problem: Determine whether the matrix A is Hermitian, where A = [[2, 1 - i], [1 + i, 3]].

Step 1: Transpose the matrix by swapping rows and columns.

A^T = \begin{pmatrix} 2 & 1+i \\ 1-i & 3 \end{pmatrix}

Step 2: Take the complex conjugate of every entry in the transposed matrix.

A^* = \overline{A^T} = \begin{pmatrix} 2 & 1-i \\ 1+i & 3 \end{pmatrix}

Step 3: Compare the result to the original matrix. Since every entry matches, A equals its conjugate transpose.

A^* = A

Answer: Yes, A is Hermitian because A = A*. Note that the diagonal entries (2 and 3) are real, and the off-diagonal entries are complex conjugates of each other.

Why It Matters

Hermitian matrices are central to quantum mechanics, where observable physical quantities (energy, momentum, spin) are represented by Hermitian operators—guaranteeing that measurement outcomes are real numbers. In numerical linear algebra and data science, Hermitian (or symmetric) matrices appear in covariance matrices and spectral decomposition, making their properties essential for principal component analysis.

Common Mistakes

Mistake: Forgetting to conjugate after transposing and checking only A = A^T.

Correction: The conjugate transpose requires both operations: transpose and then complex conjugate. Checking only the transpose misses the conjugation step, which matters whenever the matrix has complex entries.