Hermitian matrix - Knowlpedia

Definition

A complex square matrix $A=(a_{ij})$ is Hermitian if

A^*=A, \qquad\text{equivalently}\qquad a_{ij}=\overline{a_{ji}},

where $A^*=\overline A^{\,T}$ is the conjugate transpose. Hermitian matrices are the matrices of self-adjoint linear operators on finite-dimensional complex inner-product spaces in orthonormal bases.

Spectral theorem

Every Hermitian matrix has real eigenvalues and is unitarily diagonalizable: there are a unitary matrix $U$ and a real diagonal matrix $D$ such that

A=UDU^*.

Conversely, every matrix of this form is Hermitian.

The special role of $2\times2$ Hermitian matrices in Lorentz geometry is treated by the Hermitian matrix model of Minkowski space.

References

Roger A. Horn and Charles R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013. DOI record. Relevant: Chapter 4.