scholarly journals Some consequences of the rank normal form of a matrix

2019 ◽  
Vol 11 (2) ◽  
pp. 380-386
Author(s):  
Sorin Rădulescu ◽  
Marius Drăgan ◽  
Mihály Bencze
Keyword(s):  
Normal Form ◽  
Rectangular Matrix ◽  
Representation Form ◽  
The Matrix ◽  
Invertible Matrices

Abstract If A is a rectangular matrix of rank r, then A may be written as PSQ where P and Q are invertible matrices and s = \left( {\matrix{ \hfill {{{\rm{I}}_{\rm{r}}}} & \hfill {\rm{O}} \cr \hfill {\rm{O}} & \hfill {\rm{O}} \cr } } \right) . This is the rank normal form of the matrix A. The purpose of this paper is to exhibit some consequences of this representation form.

scholarly journals On the structure of least common multiple matrices from some class of matrices

2018 ◽  
Vol 10 (1) ◽  
pp. 179-184
Author(s):  
A.M. Romaniv
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Smith Normal Form ◽  
Least Common Multiple ◽  
The Matrix ◽  
Singular Matrices ◽  
Class Of Matrices ◽  
Smith Normal Forms ◽  
Invertible Matrices ◽  
Principal Ideal Domains

For non-singular matrices with some restrictions, we establish the relationships between Smith normal forms and transforming matrices (a invertible matrices that transform the matrix to its Smith normal form) of two matrices with corresponding matrices of their least common right multiple over a commutative principal ideal domains. Thus, for such a class of matrices, given answer to the well-known task of M. Newman. Moreover, for such matrices, received a new method for finding their least common right multiple which is based on the search for its Smith normal form and transforming matrices.

Arithmetic of matrices over rings

2021 ◽  
Author(s):  
V.P. Shchedryk ◽  
Keyword(s):  
Graduate Students ◽  
Normal Form ◽  
Matrix Factorization ◽  
Ring Theory ◽  
Stable Range ◽  
Matrix Rings ◽  
Principal Ideals ◽  
Factorization Theory ◽  
Close Relationship ◽  
The Matrix

The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals do- mains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a close relationship between the matrix factorization and specific properties of subgroups of the complete linear group and the special normal form of matrices with respect to unilateral equivalence. The properties of matrices over rings of stable range 1.5 are thoroughly studied. The book is intended for experts in the ring theory and linear algebra, senior and post-graduate students.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

scholarly journals Common Factors in Fraction-Free Matrix Decompositions

2020 ◽  
Author(s):  
Johannes Middeke ◽  
David J. Jeffrey ◽  
Christoph Koutschan
Keyword(s):  
Normal Form ◽  
Common Factors ◽  
Reduction Process ◽  
Triangular Matrices ◽  
Specific Data ◽  
Matrix Decompositions ◽  
The Matrix ◽  
The Common

AbstractWe consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

scholarly journals REVERSIBLE COLLAPSE OF RABBIT EARS AFTER INTRAVENOUS PAPAIN, AND PREVENTION OF RECOVERY BY CORTISONE

1956 ◽  
Vol 104 (2) ◽  
pp. 245-252 ◽  
Author(s):  
Lewis Thomas
Keyword(s):  
Normal Form ◽  
Dry Weight ◽  
Arterial Circulation ◽  
Considerable Portion ◽  
The Matrix ◽  
Ear Cartilage ◽  
Normal Shape ◽  
Trachea And Bronchi ◽  
Rabbit Ears ◽  
Complete Collapse

A substance has been demonstrated in solutions of crude papain, which, when injected intravenously into 1 kilo rabbits, in amounts less than 5 mg., results in complete collapse of both ears. The phenomenon becomes visible 4 hours after injection, and is complete within 24 hours. 3 or 4 days after papain, the ears gradually reassume their normal form. Ear collapse is associated with depletion of the ear cartilage matrix, and the disappearance of basophilia from the matrix. Similar changes occur in all other cartilage tissues, including bones, joints, larynx, trachea, and bronchi. At the time when the ears are restored to normal shape, the basophilic matrix reappears in cartilage. Repeated injections of papain, over a period of 2 or 3 weeks, bring about immunity to the phenomenon of ear collapse. When the arterial circulation to one ear is occluded for 15 minutes at the time of injection of papain, this ear is protected against collapse. The effect of crude papain could not be reproduced by crystalline papain protease or crystalline papain lysozyme, which together comprise a considerable portion of the dry weight of papain. The nature of the responsible factor has not been determined, and the possibility that chymopapain may be implicated is currently under study. Cortisone prevents the return of papain-collapsed ears to their normal shape and rigidity. Possibly this reflects a capacity of cortisone to impede the synthesis or deposition of sulfated mucopolysaccharides in tissues.

On a class of evolution algebras of "chicken" population

2014 ◽  
Vol 25 (08) ◽  
pp. 1450073 ◽  
Author(s):  
A. Dzhumadil'daev ◽  
B. A. Omirov ◽  
U. A. Rozikov
Keyword(s):  
Three Dimensional ◽  
Jordan Form ◽  
Simple Description ◽  
Rectangular Matrix ◽  
Chicken Population ◽  
The Matrix

This paper is devoted to the description of structure of evolution algebras of "chicken" population (EACP). Such an algebra is determined by a rectangular matrix of structural constants. Using the Jordan form of the matrix of structural constants we obtain a simple description of complex EACP. We give the classification of three-dimensional complex EACP. Moreover, some (n + 1)-dimensional EACP are described.

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

scholarly journals Perturbation analysis of a matrix differential equation ẋ = ABx

2018 ◽  
Vol 3 (1) ◽  
pp. 97-104 ◽  
Author(s):  
M. Isabel García-Planas ◽  
Tetiana Klymchuk
Keyword(s):  
Normal Form ◽  
Equivalence Transformations ◽  
First Order ◽  
Similarity Transformations ◽  
Canonical Pair ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Contragredient Equivalence ◽  
Canonical Matrix

AbstractTwo complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

Sign in / Sign up

Export Citation Format

Share Document