Complement de Schur i identitats detereminantals

Abstract

[en] In the first major block of this paper, the aim is to establish a clear definition of the Schur complement of a non-singular main submatrix of a block matrix, as well as some immediate basic properties that allow us to see their applications in various fields of mathematics, following the scheme proposed in [10]. Subsequently, it is intended to give a formal definition of a determinantal identity after having considered some relevant ones, seen in [4], which derive from the properties already analyzed of the Schur complement, and to then study determinantal identities extension methods with which we shall, given a starting determinant identity, get new ones. Here we highlight Muir’s Law of Extensible Minors and the Law of Cayley of the Complementarities. Finally, we will see how virtually all determinantal identities and extension laws are equivalent to each other, as shown in [6], [7] and [5].