Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

• Symmetric quadratic operators are useful models for many physical applications.
• Any such operator exhibits a pseudo-Hermitian matrix representation.
• Its eigenvalues are the natural frequencies of the Hamiltonian operator.
• The eigenvalues may be real or complex and describe a phase transition.

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit.