Subespacios hiperinvariantes y característicosuna aproximación geométrica

Resumen

The aim of this thesis is to study the hyperinvariant and characteristic subspaces of a matrix, or equivalently, of an endomorphism of a finite dimensional vector space. We restrict ourselves to the case of matrices A with an splitting characteristic polynomial, leaving for future work the generalization for any characteristic polynomial. The subspaces A-hyperinvariant and A-characteristic are subclasses of A-invariant subspaces (those containing its image for A), a key concept in the theory of matrices. Specifically, the subspaces A-hiperinvariant are those that are also invariant for all matrices that commute with A, while the A characteristic are required that are only invariant for invertible matrices that commute with A. Both concepts first appeared in the mid-30s within the context of group theory. But it was not until the 70s that appears a characterization of the A-hiperinvariant subspaces and their lattice was described in the context of matrix theory. In 2009 appears an article of Astuti and Wimmer which shows that A-hyperinvariant and A-caracteristic subspaces are the same except in the field GF(2) . In this case, Shoda theorem gives necessary and sufficient conditions for the existence of characteristic non-hiperinvariant subspaces. But the description of these subspaces was an open problem which is solved in this thesis. Our first objective, therefore, is to analyze the behavior of the centralizer of a matrix (i.e., the set of matrices commuting with it), we will assume canonical form ( Jordan or Weyr). Specifically, we calculate the determinant of the matrices in the centralizer, which in particular allows to characterize the nonsingular. Furthermore, we determined the images of a given subspace respect to the set of all matrices of these centralizers, a result that will be key for further study of hiperinvariant subspaces. We begin this study, giving conditions for the existence of one-dimensional hiperinvariant subspaces. More generally, using the results mentioned in the preceding paragraph, characterized the d-dimensional hyperinvariant subspaces associating to it a trivial Weyr partitions, which in turn allows for easy proof for the associated with certain known Segre partitions (call " hipertuplas''). These characterizations will allow us to explicitly the hiperinvariant subspaces of a given dimension, corresponding to hipertuplas with some fixed coefficient, the latter will be used in the last chapter. In the last part of the thesis, we address to study characteristic non hyperinvariant subspaces, when exist (results of Astuti-Wimmer and Shoda already mentioned). Specifically we give an explicit construction from a type of tuples associated with certain subpartitions of Segre characteristic that call "chartuplas '': to associate each two kinds of subspaces, such that the subspaces are characteristic non-hyperinvariant are precisely direct sums of two of them, one for each class. Finally, from this construction we develop an algorithm to count the number of characteristic non hiperinvariant subspaces.