Suppose I have two quadratic forms on $\mathbb R^n$, represented as symmetric matrices $A$ and $B$ on the usual basis. I am interested in approximating the function $x \mapsto \max(x^TAx, x^TBx)$ while remaining within the space of quadratic forms.
Is there a nice way to define a "maximum" operation on symmetric matrices, such that $C = \max(A,B)$ if $C$ is, in some sense, the "smallest" symmetric matrix satisfying $x^TCx \ge x^TAx$ and $x^TCx \ge x^TBx$ for all vectors $x$?
I've purposely left the notion of the "smallest" matrix $C$ undefined, as I'll accept any formalization that allows its solution to be elegantly expressed and/or easily computed. One possibility is minimizing the trace of $C$. Another, if we restrict ourselves to positive semidefinite matrices, is minimizing a convenient matrix norm.
In any case, $\max$ certainly must be commutative, and must satisfy $\max(A,A) = A$. Also, if $A$ and $B$ share the same eigenvectors, with eigenvalues $\lambda_i$ and $\mu_i$ respectively, then it seems natural that $\max(A,B)$ should also have eigenvectors the same, and eigenvalues $\max(\lambda_i,\mu_i)$. Beyond that, I can't really tell.