Let $SPD_2(\mathbb{R}^n)$ be the set of all symmetric and positive semidefinite bilinear forms $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$. Define a binary operation $(f,g) \mapsto f \ast g$ on $SPD_2(\mathbb{R}^n)$ as follows:
- pick an orthonormal basis $B$ of $\mathbb{R}^n$ and represent $f, g$ as matrices $F, G$ w.r.t. $B$;
- lef $f \ast g$ be the bilinear form associated with $F^{\frac{1}{2}}GF^{\frac{1}{2}}$ w.r.t. $B$ (it is a symmetric and positive semidefinite form).
The above operation is well-defined because it does not depend on the choice of $B$ since, for any orthogonal matrix $P$ (a change from one orthonormal basis to another), we have
$$(P^{\top}F^{\frac{1}{2}}P)(P^{\top}GP)(P^{\top}F^{\frac{1}{2}}P) = P^{\top}(F^{\frac{1}{2}}GF^{\frac{1}{2}})P$$
Is it possible to define $f \ast g$ without reference to matrix representation?
The problem for me is that matrix multiplication does not correspond to an operation on bilinear forms via isomorphism with matrices (unlike the case of the isomorphism between linear maps and matrices).