Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex 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,^[1] 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.