[dataflow]

Thanks! But I think you might've missed a subtlety here:

> This should generalize easily to the complex/Hermitian case.

This doesn't seem to be true, in that it's actually impossible to have a non-Hermitian matrix C such that x†Cx > 0 over the complex numbers for all x. Whereas over the real numbers, with a matrix R, you can have x'Rx > 0 such that R is asymmetric.

The subtlety here is that x itself can be complex in the complex case, which further constraints C to be Hermitian - see the Wikipedia link I posted above.

In other words, "complex definiteness" is actually a stronger condition than "real definiteness", even for matrices without an imaginary part.

[necroforest]

>it's actually impossible to have a non-Hermitian matrix C such that x†Cx > 0

Nice catch, it's been a few years since I had to think about these details.

[hgomersall]

I spent ages looking for a proof of the complex cases during my PhD. Most proofs of CG begin "assuming a real, positive definite matrix".

[dataflow]

Yeah, the complex case seems to get ignored in many areas. I'm guessing it's not particularly useful for many applications.