Quadratics vs Hermitian forms

Question

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!

Answer 1

For $Q$ to be a quadratic form $A$ has to be a symmetric matrix, and for $H$ to be a Hermitian form, $A$ has to be Hermitian i.e. $A = \overline{A^T}$. They are the same over real numbers, but if you work over the complex numbers of course they would be different.

An important reason why we usually use Hermitian forms when working with complex numbers is that $H(x)$ is always a real number, whereas $Q(x)$ might not be, so $\sqrt{H(x})$ can define a norm on your vector space, hence giving it a topology, so that you can discuss concepts such as continuity.

Answer 2

"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.