In this talk we investigate some classes of structures that are preserved by applying a QR step on a matrix A. We will handle two classes of such structures: the first we call polynomial structures, for example a matrix being Hermitian or Hermitian up to a rank one correction, and the second we call rank structures, which are encountered for example in all kinds of what we could call Hessenberg...

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle T ∗G of a 2n-dimensional Lie group G, which are left invariant with respect to the Lie group structure on T ∗G induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on G. Using this correspondence and results of [8] and [10], it ...

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be case for polynomial matrices expressed in monomial basis. If has a particular self-conjugate structure, show how preserve it. The structures are considered Hermitian and skew-Hermitian with respect real line, para-Herm...

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of (J2 = ±1)-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among con...