On common monic divisors having a given canonical diagonal form for matrix polynomials

1996 ◽  
Vol 79 (6) ◽  
pp. 1402-1405
Author(s):  
V. M. Petrichkovich ◽  
V. M. Prokip ◽  
F. A. Prukhnits'kii
Keyword(s):  
Diagonal Form ◽  
Matrix Polynomials ◽  
Canonical Diagonal Form

scholarly journals Petrographic structures: Khibiny ijolites and urtites

Vestnik MGTU ◽  
2021 ◽  
Vol 24 (2) ◽  
pp. 160-167
Author(s):  
Yuri Leonidovich Voytekhovsky ◽  
Alena Alexandrovna Zakharova
Keyword(s):  
Crystalline Rocks ◽  
Diagonal Form ◽  
Standard Description ◽  
Khibiny Massif ◽  
Constitutional Principle ◽  
Hardy Weinberg Equilibrium ◽  
Definition Of ◽  
Canonical Diagonal Form ◽  
Strict Definition

In addition to the standard description of the structures and textures of crystalline rocks the mathematical approaches have been proposed based on a rigorous determination of the petrographic structure through the probabilities of binary intergrain contacts. In general, the petrographic structure is defined as an invariant aspect of rock organization, algebraically expressed by the canonical diagonal form of the symmetric Pij matrix and geometrically visualized by structural indicatrices - surfaces of the 2nd order. The agreed nomenclature of possible petrographic structures for an n-mineral rock is simple: the symbol Snm means that there are exactly m positive numbers in the canonical diagonal form of the Pij matrix. New types of barycentric diagrams have been proposed. To describe the massive texture, the concept of Hardy - Weinberg equilibrium has been proposed. This boundary classifies barycentric diagrams into areas within which canonical types of Рij matrices and topological types of structural indicatrices are preserved. The change in the organization of the rock within a type is quantitative, the transition from one type to another means structural restructuring. The methods are used to describe ijolites and urtites of the Khibiny massif, the Kola Peninsula. In the modern taxonomy of rocks, the boundaries between them are mostly conditional and are drawn according to the contents of rock-forming minerals, for example, between ijolites and urtites - according to the contents of nepheline and pyroxene. The strict definition of the petrographic structure proposed by the authors makes it possible to introduce into petrography the constitutional principle (structure + composition), which is successfully acting in mineralogy.

scholarly journals Petrographic structures and Hardy – Weinberg equilibrium

2020 ◽  
Vol 242 ◽  
pp. 133
Author(s):  
Yury VOYTEKHOVSKY ◽  
Alena ZAKHAROVA
Keyword(s):  
Quadratic Forms ◽  
Dimensional Space ◽  
Complete Classification ◽  
Algebraic Expression ◽  
Diagonal Form ◽  
Hardy Weinberg Equilibrium ◽  
Algebraic Forms ◽  
Canonical Diagonal Form

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data.

On relations betweenR a Rb transformation and canonical diagonal form of λ-matrix

1997 ◽  
Vol 40 (3) ◽  
pp. 258-268 ◽  
Author(s):  
Renji Tao ◽  
Peirong Feng
Keyword(s):  
Diagonal Form ◽  
Canonical Diagonal Form

Smith form and Kronecker canonical form

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Form ◽  
Matrix Polynomial ◽  
Diagonal Form ◽  
Quaternion Matrix ◽  
Real Matrix ◽  
Complete Proof ◽  
Matrix Polynomials ◽  
Unimodular Matrix ◽  
Complex Coefficients ◽  
Kronecker Canonical Form

This chapter treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form), which asserts that every quaternion matrix polynomial can be brought to a diagonal form under pre- and postmultiplication by unimodular matrix polynomials, is proved for such polynomials. In contrast to matrix polynomials with real or complex coefficients, a Smith form is generally not unique. For matrix polynomials of first degree, a Kronecker form—the canonical form under strict equivalence—is available, which this chapter presents with a complete proof. Furthermore, the chapter gives a comparison for the Kronecker forms of complex or real matrix polynomials with the Kronecker forms of such matrix polynomials under strict equivalence using quaternion matrices.

Sufficient conditions for Hurwitz and Schur stabilty of interval matrix polynomials

1999 ◽  
Author(s):  
Yang Xiao ◽  
Rolf Unbehauen ◽  
Xiyu Du
Keyword(s):  
Sufficient Conditions ◽  
Interval Matrix ◽  
Matrix Polynomials

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young
Keyword(s):  
Unit Circle ◽  
Matrix Polynomial ◽  
Hermitian Matrix ◽  
Krein Space ◽  
Gram Matrix ◽  
Natural Basis ◽  
Test Matrix ◽  
Matrix Polynomials ◽  
Rational Vector ◽  
Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

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