Canonical Diagonal Form Research Articles

A Bezout ring with nonzero principal Jacobson radical

In this paper, we study a commutative Bezout domain with nonzero Jacobson radical being a principal ideal. It has been proved that such a Bezout domain is a ring of the stable range 1. As a result, we have obtained that such a Bezout domain is a ring over which any matrix can be reduced to a canonical diagonal form by means of elementary transformations of its rows and columns.

Methodology for determination of petrographic structures using the MIU-5M device

The authors develop a statistical method for determining the structures of rocks. A mathematically rigorous definition of the petrographic structure through the probabilities of binary intergranular contacts is proposed. The petrographic structure is defined as an invariant aspect of rock organization algebraically expressed by the canonical diagonal form of symmetric matrix of contact probabilities and visualized by structural indicatrices - 2 order surfaces. A consistent nomenclature of petrographic structures has been proposed. Earlier the method showed itself well in describing the structures of ijolites and urtites of the Khibiny Mountains, as well as amphibolites of the islands of Keretsky Archipelago (the White Sea). Mass application of the method is restrained by the routine procedure of calculating the probabilities of intergranular contacts in petrographic thin sections under a microscope. The problem of modern optical structure analyzers is assignment different sections of the same anisotropic mineral to different minerals, and separation blocks of a mineral individual to different individuals. The paper describes the experience of using the domestic MIU-5M device for scanning petrographic thin sections by the linear method of Rosival. The combination of visual control and minimum automation of the process allows us to achieve convergence of the structural types with results obtained by manual processing of image of petrographic thin section. Additionally, thin sections have been studied, for which the structural types differ with various methods of counting. It has been revealed that the presence of poikilitic ingrowths and features of the mineral composition (the existence of hypergene changes or minerals with similar optical properties) affect the statistics of probabilities. Recommendations are given on the selection of the scanning step depending on the grain size in the studied rock.

Petrographic structures: Khibiny ijolites and urtites

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.

Petrographic structures and Hardy – Weinberg equilibrium

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.

Monic divisors with one invariant factor of polynomial matrices

The problem of separation of a regular factor from a matrix polynomial has been of interest to mathematicians for a long time because the solution of this problem gives a method for the determination of roots of matrix one-sided equations. Various methods of contemporary mathematics are used for the solution of this problem. In 1956, Lopatyns’kyi, in terms of systems of differential equations, gave necessary and sufficient conditions for representation of a matrix in the form of the product of regular factors with coprime determinants. Gochberg, Lancaster, and Rodman in [7] and Malyshev in [4] solve this problem by the methods of Jordan chains. In [3], Kazimirs’kyi proposes an essentially new approach to the problem of separation of a regular divisor from a matrix polynomial. This approach is based on the notions of a determining matrix and values of a matrix on a system of roots of a polynomial. In these terms, Kazimirs’kyi established necessary and sufficient conditions for the existence of these divisors and proposes a method for their determination. The obtained results hold for polynomial matrices over an algebraically closed field of characteristic zero. The specific feature of methods of solution and terms used in the representation of results do not enable one to generalize these methods for wider classes of fields. This generalization became possible after work [2] in which the Kazimirs’kyi results are represented in another form. In the present paper, we investigate polynomial matrices over an infinite field. Under certain restrictions on the canonical diagonal form of a divisor, necessary and sufficient conditions for its existence are established. It is worth noting that, after the solution of the problem of separation of a regular divisor by Kazimirs’kyi, the number of works devoted to regular divisors sharply dropped. Problems of description of nonassociative divisors of matrices [6, 10] and representation of matrices in the form of the product of several factors with preassigned characteristics [5, 8, 13] have attracted the attention of researchers. Let F be a field and let A(x) be a nonsingular (n × n) matrix over F[x] representable in the form of a matrix polynomial over the field F : A(x) = A k x k + A k−1 x k−1 +… + A 0 . The matrix A(x) is called monic if A k = E is an identity matrix of order k and regular if det A k ≠ 0. We say that the matrix A(x) is right regularizable if there exists an invertible matrix U (x) such that

Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems

We consider the initial-boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes. The switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time. We show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching. We also provide an explicit dwell-time bound for guaranteeing exponential stability of the switching system when, for each mode, the system is exponentially stable. Our stability conditions only depend on the system parameters and boundary data. These conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption. We present tutorial examples to illustrate the instabilities that can result from switching.

On interdependence between invariant factors of a block-triangular matrix and its diagonal blocks

The interdependence between the invariant factors of a block-triangular matrix and its diagonal blocks is established under some constraints on its canonical diagonal form.

Transforming matrices and divisors generated by them

Over a commutative domain of elementary divisors, each matrix is a product of invertible matrices and a certain diagonal matrix, which are called, respectively, transforming matrices and a canonical diagonal form. We establish necessary and sufficient conditions when all divisors of the matrix that have a canonical diagonal form given beforehand are described using only transforming matrices.

On a canonical form of a 3×3 Hermitian matrix over the ring of integral split octonions

We show that a 3×3 Hermitian matrix over the ring of integral split octonions has a unique canonical diagonal form under the action of the group E6,6(Z). This confirms the conjecture of J. Maldacena, G. Moore, and A. Strominger [hep-th/9903163] on a standard form of a charge vector.

On Multiplicativity of Canonical Diagonal Forms of Matrices over the Domain of Principal Ideals. II

We investigate the structure of matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms.

On the parallel factorization of matrices over principal ideal rings

We describe all the factorizations A=BC (up to associates) of a matrix A over a commutative principal ideal domain parallel to the factorization DA=Φψ of its canonical diagonal form DA (Φ and ψ are diagonal matrices), that is, the factorizations such that the matrices B and C are equivalent to the matrices Φ and ψ respectively.

On a class of divisors of matrices

We propose a structural criterion for divisibility of matrices over a commutative principal ideal domain. Under certain restrictions we exhibit a method of finding all the non-associate divisors that have a prescribed canonical diagonal form.

Diagonalization of an element P of $\mathfrak{P}^{c}$ by the compact Lie group $E_{7}$

The following diagonalization problems are well-known. A Hermitian matrix X e M(n,K) over K = R,C,H can be transformed to a diagonal form by some element A of the groups SO(n), SU(n), Sp(n), respectively.Moreover we know that a Hermitian matrix X e M(3,K) over K = (£,(Ec can be transformed to a diagonal form by some element a of the compact Lie groups F4, E&, respectively (Freudenthal [1],Yokota [9]).Now, in this paper, we show that an element P of ^5C (which is a minimal dimensional representation space of the compact Lie group E-j) can be transformed to a diagonal form by some element a of £7 (Theorem 5).In the last section, we give a canonical diagonal form of P of ^pc (Theorem 12). We would like to thank T. Miyashita for his calculation of Lemma 3.

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D A and D B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V A, and V B over R such that UAV a=DA and UAV B=DB if and only if the matrices B *A and D * B DA where B * 0 is the matrix adjoint to B, are equivalent.

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

R aRb transformation over λ-matrix is defined and explored. Relations betweenR aRb transformation over λ-matrix and canonical diagonal form of λ-matrix are investigated and some results are given.

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

Under certain restrictions we give a condition for the existence of common monic divisors having a given diagonal form for nonsingular matrix polynomials, and we propose a method of finding them.

To the problem of multiplicativity of canonical diagonal forms of matrices over the domain of principal ideals

We study the structure of nonsingular matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms. In particular, we establish necessary and sufficient conditions of multiplicativity of canonical diagonal forms of nonsingular matrices over this domain.

On the uniqueness of the unital divisor of a matrix polynomial over an arbitrary field

Let P be an arbitrary field and let P(x) be a ring of polynomials over P. Denote by Pn and Pn (x) the rings of n xn matrices over P and P(x), respectively. Consider a nonsingular polynomial matrix A(x) e P~ (x) (detA(x) $ 0) and represent it as a matrix polynomial over the field P: is a unital matrix polynomial. In this paper, we establish conditions under which a unital divisor D (x) extracted from the matrix polynomial A (x) is determined uniquely by its characteristic polynomial detA (x) = d(x). The re- sult obtained enables one to indicate the class of matrix polynomials for which this unital divisor is unique for a given characteristic polynomial. We also show how this result can be used when solving matrix polynomial equations. The uniqueness of a tmital divisor for a matrix polynomial over an algebraically closed field with char- acteristic zero was studied in ( 1 ). It is known (2) that, for a matrix A (x) e Pn (x), there exist matrices U(x), V(x) ~ GL n (Pn (x)) such that U (x)A (x) V(x) = F A (X) = diag (al(x), a2(x) ..... an(x)), where aj(x) e P(x), j= 1, 2 ..... n, are unital polynomials and ai(x ) (ai+l(x), i = 1, 2 ..... n - 1. The matrix FA(X ) is called a canonical diagonal form of the matrix A(x). It is also known that if a matrix is nonsingular, then it can be reduced to the normal Hermite form by right elementary transformations, i.e., for the matrix A (x), there exists a matrix W(x) e GLn (P(x)) such that

Decomposition of polynomial matrices into factors with prescribed canonical diagonal forms

We give a test for decomposability of a polynomial matrix over an arbitrary infinite field into factors with prescribed canonical diagonal forms whose product is the canonical diagonal form of the given matrix. We exhibit a method of actually constructing such decompositions of polynomial matrices.

Parallel factorizations of polynomial matrices

Conditions are established under which suggested factorizations of polynomial matrices over a field are parallel to factorizations of their canonical diagonal forms. An existence criterion of these factorizations of polynomial matrices is indicated and a method of constructing them is suggested.