Conjugate gradient methods for real symmetric systems can be applied to complex Hermitian systems in a straightforward manner. For non-Hermitian complex systems we distinguish two cases. In general, for any coefficient matrix a CGNE method is possible, that is, a conjugate gradients method on the normal equations , or one can split the system into real and complex parts and use a method such as GMRES on the resulting real nonsymmetric system. However, in certain practical situations the complex system is non-Hermitian but symmetric.
Complex symmetric systems can be solved by a classical conjugate gradient or Lanczos method, that is, with short recurrences, if the complex inner product is replaced by . Like the BiConjugate Gradient method, this method is susceptible to breakdown, that is, it can happen that for . A look-ahead strategy can remedy this in most cases (see Freund  and Van der Vorst and Melissen ).