Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of a partial solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2. We also give Cramer’s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well

Determinantal Representations of the Core Inverse and Its Generalizations

Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author

Explicit Determinantal Representation Formulas of W

By using determinantal representations of the W-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equations WAWX=D, XWBW=D, and W1AW1XW2BW2=D

Parametrized solutions X of the system AXA = AEA and A^k EAX = XAEA^k for a matrix A having index k

[EN] Let A and E be n x n given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by AXA = AEA and A(k) EAX = XAEA(k) , for k being the index of A. In addition, its general solution is derived in terms of a G-Drazin inverse of A. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas.Partially supported by Universidad Nacional de Rio Cuarto (grant PPI 18/C472) and CONICET (grant PIP 112-201501-00433CO). Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. Nro. 155/14). Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (grant 24/L108).Ferreyra, DE.; Latanzi, M.; Levis, F.; Thome, N. (2019). Parametrized solutions X of the system AXA = AEA and A^k EAX = XAEA^k for a matrix A having index k. The Electronic Journal of Linear Algebra. 35:503-510. https://doi.org/10.13001/1081-3810.4051S5035103

Algebraic Legendrian Varieties

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover its action preserves the contact structure. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The contribution of this thesis is in three steps: First we find an example of a smooth toric surface. Next we find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding. Finally, we realise that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. By applying this result to known examples and decomposable Legendrian varieties, we construct infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants.Comment: 116 pages, 6 figures. Author's PhD thesis (corrected and improved), defended on Feb 7th, 2008. to appear in Dissertationnes Mathematica