A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to its own complex conjugate when the transpose operation is applied.
Some properties of a Hermitian matrix include: all eigenvalues are real, all diagonal elements are real, and the matrix is self-adjoint.
In quantum mechanics, Hermitian matrices are used to represent observables such as energy and angular momentum. The eigenvalues and eigenvectors of these matrices correspond to the possible values and states of the observable.
A Hermitian matrix is a special case of a unitary matrix, where the matrix is also equal to its inverse. This means that the matrix is both self-adjoint and unitary.
No, a non-square matrix cannot be Hermitian. A Hermitian matrix must be square in order for the transpose and complex conjugate operations to be applied. If the matrix is not square, these operations cannot be performed and the matrix cannot be considered Hermitian.