Quadratic Forms and Associated Matrices

Question

This might be a dumb question but when we write the matrix associated with a quadratic form, why does the matrix need to be symmetric in general? I'm asking because I'm thinking there isn't a unique matrix that is associated with a quadratic form unless you specifically state that the matrix has to be symmetric. If it has to be symmetric, why do some textbooks do not note that the matrix associated with the quadratic form needs to be symmetric?

Answer 1

Correct. The matrix used in a quadratic form does not need to be symmetric. For instance, $x^T Q x$ is a quadratic form for any matrix $Q$. The key part here is that we can always choose the matrix to be symmetric. In particular, for $Q' = (Q + Q^T)/2$, we have

$x^T Q' x = (1/2) x^T Q x + (1/2) x^T Q^T x = x^T Q x$

Notice the equality here is because $x^T Q^T x = (x^T Q^T x)^T = x^T Q x$.

Therefore, we always assume that the matrix defining a quadratic form is symmetric.

Symmetry has many nice properties, so it is convenient to just start with the symmetric version.

Answer 2

It's part of the definition of a quadratic form: it must derive from a bilinear symmetric form.

In detail: According to Bourbaki, Algebra, ch.9, a quadratic form on an $A$-module $E$ ($A$ commutative) is a map $Q\colon E\to A$ such that:

(i) $Q(\alpha x)=\alpha^2 Q(x)$ for all $\alpha\in A$ and $x\in E$

(ii) the map $\;\begin{aligned}[t]E \times E&\to A\\(x,y)&\mapsto \Phi(x,y)=P(x+y)-P(x)-P(y) \end{aligned}$ is a bilinear form (necessarily symmetric).