Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse

Characterization and Representation of Weighted Core Inverse of Matrices

In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are explored. Through one-sided core and dual-core inverse, the existence of a generalized weighted Moore-Penrose inverse of matrices is proposed. Further, by applying a new representation and using the properties of the weighted core inverse of a matrix, we discuss a few new results related to the reverse order law for these inverses.Comment: 18 page

Junction type representations of the Temperley-Lieb algebra and associated symmetries

Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang-Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetry is studied.Comment: 19 pages, LaTex. Published in Symmetry, Integrability and Geometry: Methods and Applications (SIGMA

Representations and symbolic computation of generalized inverses over fields

This paper investigates representations of outer matrix inverses with prescribed range and/or none space in terms of inner inverses. Further, required inner inverses are computed as solutions of appropriate linear matrix equations (LME). In this way, algorithms for computing outer inverses are derived using solutions of appropriately defined LME. Using symbolic solutions to these matrix equations it is possible to derive corresponding algorithms in appropriate computer algebra systems. In addition, we give sufficient conditions to ensure the proper specialization of the presented representations. As a consequence, we derive algorithms to deal with outer inverses with prescribed range and/or none space and with meromorphic functional entries.Agencia Estatal de investigaciónUniversidad de Alcal

Representations and geometrical properties of generalized inverses over fields

In this paper, as a generalization of Urquhart’s formulas, we present a full description of the sets of inner inverses and (B, C)-inverses over an arbitrary field. In addition, identifying the matrix vector space with an affine space, we analyze geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of (B, C)-inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.Agencia Estatal de InvestigaciónUniversidad de Alcal

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture