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Relationship between 2-normal and binormal matrices

Investigate whether the class of 2-normal matrices—those for which the coval expansion det p(A) contains only the first two terms—has any relationship to the class of binormal matrices defined by the commutativity of A^*A and AA^*, and ascertain whether there is equivalence, inclusion, or any structural connection between these two notions.

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Background

The paper introduces a "greedy" coval expansion for det p(A) and defines a matrix to be 2-normal if only the first two terms are present. Geometrically, such matrices admit a particularly simple coval description in terms of eigenvalues and secondary values.

The author points to the existing notion of binormal matrices—where A*A and AA* commute—and explicitly notes uncertainty about any connection between this classical class and the newly introduced 2-normal class.

References

In the literature, there exists so-called binormal matrices -- see [Ikramov]. A matrix A is binormal if A\star A, A A\star commute. At this moment, I do not know if the two concepts are related or not.

— Coval description of the boundary of a numerical range and the secondary values of a matrix (2410.03744 - Blaschke, 1 Oct 2024) in Section 6 (Open problems), Subsection “Properties of a '2-normal' matrix”