Quadratics vs Hermitian forms The quadratic form is given by:
$$Q(\mathbf{x}) =\langle\ \mathbf{x} \ | \ A\mathbf{x} \rangle = x^TAx$$
Where $Q$ is a real scalar and hence $Q = Q^T$
The hermitian form is given by:
$$H(\mathbf{x}) = \langle \ \mathbf{x} \ | \ A\mathbf{x} \ \rangle=x^{\dagger}Ax$$
Where $H$ is also scalar and hence $H = H^T$
Now I'm new to the subject and these 2 are really similar and I get quite confused when to use which since both can be used with complex numbers.
Under exactly what circumstances do we use the Hermitian form and when do we use quadratic?
Thanks!

"When to use which": that depends on your application, I suppose. Almost all quadratic forms that I encounter (in differential geometry and physics) encode inner products, and so are positive-definite, and symmetric for real vector spaces and Hermitian for complex ones. It would be very odd for me to encounter a symmetric, but non-Hermitian, complex quadratic form, but perhaps there are other areas of mathematics where these are more common.
