# When will the Orthomin/CG iteration fails

I know that the the Conjugate Gradient iteration fails when $$0\in \mathcal {W}(A^{H})$$, which means there's a complex vector $$x+iy$$ such that $$(x+iy)^{T}A^{H}(x+iy)=0$$. I wonder how to derive a real initial vector such that the Conjugate Gradient iteration fails if $$A$$ and $$b$$ are real based on this complex vector? The thing is that I don't know where I can use the condition $$b$$ is real.

The equivalent condition for the real case is that the iteration will fail if there is a vector $$x\in{\mathbb R}^n$$ so that $$x^T A x = 0$$, which is equivalent to saying "the iteration will fail if the matrix $$A$$ has a zero eigenvalue". Or, equivalently "the iteration will fail if the matrix $$A$$ has a null space".
• If the matrix has a null space but the initial residual does not have a component in the null space, then the iteration will give you a solution of the linear system $$Ax=b$$. This problem does not have a unique solution if $$A$$ has a null space, but you then simply get that solution that is perpendicular to the null space -- which happens to also be the least-squares solution of the underdetermined problem (or what you would get with the pseudo-inverse, $$x=A^\dagger b$$).