As discussed in §9.2.2, one can use either the generalized eigenvalue problem (9.4) with (9.5) or the generalized eigenvalue problem (9.9) with (9.10) for solving the corresponding quadratic eigenvalue problem (9.1).
If all matrices , , and are Hermitian and is positive definite, as in the special case (9.2), then the decision comes down to choosing either intrinsically non-Hermitian generalized eigenvalue problem (9.4) and (9.5), with a Hermitian positive definite matrix, or a generalized eigenvalue problem (9.9) and (9.10), where both and matrices are Hermitian but neither of them will be positive definite.
Numerical methods discussed in Chapter 8 can be used for solving these generalized ``linear'' eigenvalue problems. For example, in the MSC/NASTRAN , the non-Hermitian formulation (9.4) with (9.5) is used.
The symmetric indefinite Lanczos method discussed in §8.6 is specifically targeted for the generalized symmetric indefinite eigenvalue problem (9.9). The potential trouble is that in the symmetric indefinite Lanczos method, the basis vectors are orthogonal with respect to an indefinite inner product. Therefore, these basis vectors may not be linearly independent and the algorithm may break down and be numerically unstable. Nevertheless, this is often an attractive way to solve the original QEP because of potential savings in memory requirements and floating point operations. See §8.6 for further details.