On the diagonalization of the general quadratic Hamiltonian for coupled harmonic oscillators

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Abstract
It is shown that the general quadratic Hamiltonian for coupled harmonic oscillators can be diagonalized, provided the matrix of the quadratic form is positive definite. This condition is also necessary if the frequencies of the resulting uncoupled oscillators are to be positive. The construction of a diagonalizing matrix follows the usual procedure as in the Hermitian case; the only difference being a change of the metric from I= (1 1) to J= (1 −1).

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