Hermitian matrices can be understood as the complex extension of real symmetric matrices.
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
This is also the way that the more general concept of self-adjoint operator is defined.
The diagonal elements must be real, as they must be their own complex conjugate.
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.