Relationship between 2-normal and binormal matrices

Ascertain whether the class of 2-normal matrices—those for which the coval expansion of det p(A) reduces to two terms, det p(A) = ∏_{j=1}^n p(λ_j) − (t_1(A)/4)∏_{j=1}^{n-2} p(χ_j)—is related to the classical class of binormal matrices (matrices A for which A* A and A A* commute), and characterize any equality, inclusion, or other structural connection between these classes.

Background

Normal matrices yield a single-term coval identity for the Kippenhahn curve. The author defines 2-normal matrices as those for which exactly two terms appear in the coval expansion, making their numerical ranges describable using eigenvalues and secondary values only.

Binormal matrices (A* A commuting with A A*) form a well-studied class in the literature. The author explicitly notes that it is unknown whether 2-normal matrices relate to binormal matrices, motivating a comparison between these two notions.