In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: $${\displaystyle A{\text{ is Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}$$

or in matrix form: $${\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.}$$

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix ${\displaystyle A}$ is denoted by ${\displaystyle A^{\mathsf {H}},}$ then the Hermitian property can be written concisely as

$${\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}$$

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are ${\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast },}$ although in quantum mechanics, ${\displaystyle A^{\ast }}$ typically means the complex conjugate only, and not the conjugate transpose.

Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:

A square matrix ${\displaystyle A}$ is Hermitian if and only if it is equal to its conjugate transpose, that is, it satisfies $${\displaystyle \langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}$$ for any pair of vectors ${\displaystyle \mathbf {v} ,\mathbf {w} ,}$ where ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the inner product operation.