The level curves of the angle function of a positive definite symmetric matrix (Q2901371)

Given a real \(n \times n\) matrix \(A\), let \(\phi_A\) be the maximum angle by which \(A\) rotates a unit vector. It is easy to see that a symmetric matrix \(A\) is positive definite iff \(\phi_A<\frac{\pi}2\). The Jordan product \((A,B)\mapsto A\circ B\) of two symmetric matrices is again a symmetric matrix. If \(A\) and \(B\) are positive definite and \(\phi_A+\phi_B<\frac{\pi}2\), then \(A\circ B\) is again positive definite. On the other hand, if \(\phi_A+\phi_B\geq\frac{\pi}2\), then there exists a special orthogonal matrix \(Q\) such that \(A\circ QBQ^{-1}\) is indefinite. The article under review concerns the following question: if \(A\) and \(B\) are commuting positive definite matrices such that \(\phi_A+\phi_B\geq\frac{\pi}2\), what is \(\inf\{\phi_Q\mid Q \in \text{SO}_n,\, A\circ QBQ^{-1}\) is indefinite\(\}\)? This problem is considered for small matrix sizes, and the level curves of the angle function \(x\mapsto \angle \langle Ax,x\rangle\) for \(A\) positive definite are studied.