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We present an intuitive and scalable algorithm for the diagonalization ofcomplex symmetric matrices, which arise from the projection ofpseudo--Hermitian and complex scaled Hamiltonians onto a suitable basis set of"trial" states. The algorithm diagonalizes complex and symmetric(non--Hermitian) matrices and is easily implemented in modern computerlanguages. It is based on generalized Householder transformations and relies oniterative similarity transformations T -> T' = Q^T T Q, where Q is a complexand orthogonal, but not unitary, matrix, i.e, Q^T equals Q^(-1) but Q^+ isdifferent from Q^(-1). We present numerical reference data to support thescalability of the algorithm. We construct the generalized Householdertransformations from the notion that the conserved scalar product ofeigenstates Psi_n and Psi_m of a pseudo-Hermitian quantum mechanicalHamiltonian can be reformulated in terms of the generalized indefinite innerproduct [integral of the product Psi_n(x,t) Psi_m(x,t) over dx], where theintegrand is locally defined, and complex conjugation is avoided. A few examplecalculations are described which illustrate the physical origin of the ideasused in the construction of the algorithm.

Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem