On the Algebraic Structure of the Moore Penrose Inverse of Full Row or Full Column Rank Polynomial Matrices

2019 ◽  
Author(s):  
Ioannis S. Kafetzis ◽  
Nicholas P. Karampetakis
Keyword(s):  
Algebraic Structure ◽  
Full Column Rank ◽  
Polynomial Matrices ◽  
Rank Polynomial

scholarly journals Stratification of full rank polynomial matrices

2013 ◽  
Vol 439 (4) ◽  
pp. 1062-1090 ◽  
Author(s):  
Stefan Johansson ◽  
Bo Kågström ◽  
Paul Van Dooren
Keyword(s):  
Full Rank ◽  
Polynomial Matrices ◽  
Rank Polynomial

scholarly journals On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems

2009 ◽  
Vol 19 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Antonis-Ioannis Vardulakis ◽  
Nicholas Karampetakis ◽  
Efstathios Antoniou ◽  
Evangelia Tictopoulou
Keyword(s):  
Transfer Function ◽  
State Space ◽  
Algebraic Structure ◽  
Space Systems ◽  
Realization Theory ◽  
Polynomial Matrices ◽  
State Space Systems ◽  
Generalized State Space ◽  
The Difference ◽  
Generalized State

On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space SystemsWe review the realization theory of polynomial (transfer function) matrices via "pure" generalized state space system models. The concept of anirreducible-at-infinitygeneralized state space realization of a polynomial matrix is defined and the mechanism of the "cancellations" of "decoupling zeros at infinity" is closely examined. The difference between the concepts ofirreducibilityandminimalityof generalized state space realizations of polynomial (transfer function) matrices is pointed out and the associated concepts ofdynamicandnon-dynamicvariables appearing in generalized state space realizations are also examined.

scholarly journals Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

scholarly journals Polymorphic Iterable Sequential Effect Systems

2021 ◽  
Vol 43 (1) ◽  
pp. 1-79
Author(s):  
Colin S. Gordon
Keyword(s):  
Algebraic Structure ◽  
Position Effect ◽  
Type Systems ◽  
Sequential Effect ◽  
Prior Work ◽  
Sequential Effects ◽  
Wide Range ◽  
The Relationship ◽  
Categorical Semantics

Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of effects is irrelevant. Understanding such systems in terms of a semilattice of effects grounds understanding of the essential issues and provides guidance when designing new effect systems. By contrast, sequential effect systems—where the order of effects is important—lack an established algebraic structure on effects. We present an abstract polymorphic effect system parameterized by an effect quantale—an algebraic structure with well-defined properties that can model the effects of a range of existing sequential effect systems. We define effect quantales, derive useful properties, and show how they cleanly model a variety of known sequential effect systems. We show that for most effect quantales, there is an induced notion of iterating a sequential effect; that for systems we consider the derived iteration agrees with the manually designed iteration operators in prior work; and that this induced notion of iteration is as precise as possible when defined. We also position effect quantales with respect to work on categorical semantics for sequential effect systems, clarifying the distinctions between these systems and our own in the course of giving a thorough survey of these frameworks. Our derived iteration construct should generalize to these semantic structures, addressing limitations of that work. Finally, we consider the relationship between sequential effects and Kleene Algebras, where the latter may be used as instances of the former.

scholarly journals Typical ranks of certain 3-tensors and absolutely full column rank tensors

2017 ◽  
Vol 66 (1) ◽  
pp. 193-205 ◽  
Author(s):  
Mitsuhiro Miyazaki ◽  
Toshio Sumi ◽  
Toshio Sakata
Keyword(s):  
Full Column Rank

scholarly journals Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Gradient Descent ◽  
Main Idea ◽  
Full Column Rank ◽  
Minimum Error ◽  
Matrix Coefficients ◽  
Special Cases ◽  
Efficiency And Effectiveness ◽  
Associated Matrix

AbstractThis paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Interpolation algorithm of Leverrier–Faddev type for polynomial matrices

2006 ◽  
Vol 42 (3-4) ◽  
pp. 345-361 ◽  
Author(s):  
Marko D. Petković ◽  
Predrag S. Stanimirović
Keyword(s):  
Interpolation Algorithm ◽  
Polynomial Matrices
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