When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices?

Abstract

The mapping Φ n (A, B) = AB − BA, where the matrices A, B ∈ C 2n×2n are skew-Hamiltonian with respect to transposition, is studied. Let C n be the range of Φ n : we give an implicit characterization of C n , obtaining results that find an application in algebraic geometry. Namely, they are used in [R. Abuaf and A. Boralevi, Orthogonal bundles and skew-Hamiltonian matrices, In Preparation] to study orthogonal vector bundles. We also give alternative and more explicit characterizations of C n for n ≤ 3. Moreover, we prove that for n ≥ 4 the complement of C n is nowhere dense in the set of 2n-dimensional Hamiltonian matrices, denoted by H n , implying that almost all matrices in H n are in C n for n ≥ 4. Finally, we show that Φ n is never surjective as a mapping from W n × W n to H n , where W n is the set of 2n-dimensional skew-Hamiltonian matrices. Along the way, we discuss the connections of this problem with several existing results in matrix theory.

Orthogonal Bundles and Skew-Hamiltonian Matrices

“…The study of the varieties C r,n has led us to prove a somewhat curious result, that we have not been able to find anywhere in literature, and that we therefore report here. Similar results were recently obtained by [Nof13] using completely different techniques.…”

“…Even more interesting would be finding out whether the map φ composed with the projection P(S 2 V ) ։ P(S 2 V ≤r ) is surjective for some value of the rank r. In other words, is there an r for which all symmetric matrices of rank r are of the form AJB − BJA for a pair of skew-symmetric matrices (A, B)? Noferini [Nof13] has shown that the statement is false for r = 2, any n. The question remains open for higher values of r.…”

“…for some coefficients a and b. d) We perform a local computation and then induction on p = n/2. A different proof of the fact that φ is dominant can be found in [Nof13,Proposition 4.4], where the author proves that for n ≥ 8 the set of symmetric matrices with all distinct eigenvalues is inside Im φ. So let us start our induction.…”