The generalized Lilbert matrix

We introduce a generalized Lilbert [Lucas-Hilbert] matrix. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger’s celebrated algorithm.     

The generalized Fibonomial matrix

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k+1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples.     

The generalized inverse of a perturbed singular matrix

It is shown that, whenA is singular, (A+A ₁)^–1 can be expanded into a Laurent's series in. The coefficients of the expansion are given in an explicit form. The case where A+A ₁ vanishes identically in is also studied and a generalized inverse ofA+A ₁ is given.

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Zusammenfassung Es wird gezeigt, daß wennA singulär ist (A+A ₁)^–1 auf eine Laurent-Serie in erweitert werden kann. Die Koeffizienten der Erweiterung werden in expliziter Form gegeben. Der Fall wo A+A ₁ sich identisch in auflöst wird ebenfalls untersucht und eine verallgemeinerte Umkehrung vonA+A ₁ wird angegeben.

     

A generalized unitary Hessenberg matrix

This paper describes the use of a generalized isometric Arnoldi algorithm to reduce a unitary matrix, via unitary similarity, to a product of elementary reflectors and permutations. The computation is analogous to the reduction of a unitary matrix to a unitary Hessenberg matrix using the isometric Arnoldi algorithm. In the case in which A is a shift matrix, the reduction provides a novel recurrence for the factor R in the QR factorization of a Toeplitz-like matrix.     

Isomorphism of generalized matrix rings

An isomorphism problem is considered for generalized matrix rings with values in a given ring R. An exhaustive answer is given for the case of a commutative domain R and a commutative local ring R.     

On certain generalized matrix norms

Two methods of generating classes of generalized matrix norms are presented. Particular norms obtained in this manner are investigated in some detail.     

The general coupled matrix equations over generalized bisymmetric matrices

In the present paper, by extending the idea of conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations

     

k-commuting mappings of generalized matrix algebras

Periodica Mathematica Hungarica - In this paper we will study k-commuting mappings of generalized matrix algebras. The general form of arbitrary k-commuting mapping of a generalized matrix algebra...     

Applications of the generalized Sylvester matrix

Recently a Sylvester matrix for several polynomials has been defined, establishing the relative primeness and the greatest common divisor of polynomials. Using this matrix, we perform qualitative analysis of several polynomials regarding the inners, the bigradients, Trudi's theorem, and the connection of inners and the Schur complement. Also it is shown how the regular greatest common divisor of m+1 (m>1) polynomial matrices can be determined.     

Properties of a generalized Sylvester matrix

We consider problems close to that of the minimal stabilization of a linear vector (i.e., MISO or SIMO) dynamic system; more specifically, the problem of determining the number of common roots of a family of polynomials, and investigating the properties of the so-called generalized Sylvester matrix. The classical definition of the Sylvester matrix is valid for two polynomials, and there are different methods for defining the generalized (extended) Sylvester matrix for a family of polynomials. In this work, we consider a definition of the generalized Sylvester matrix and its properties in the context of their potential future application for solving the minimal stabilization problem.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

Net subrings of generalized matrix rings

Let Λ be a ring with the following properties: (a) Λ is a direct sum of left ideals P₁,..., P_n; (b) every nontrivial homomorphism P_i→P_j is a monomorphism; (c) for every i, j the intersection of any two submodules of P_j isomorphic to P_i contains a submodule isomorphic to P_i. Then Λ can be represented as a subring associated with a net of ideals in a generalized matrix ring. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 184–198. Translated by A. V. Yakovlev.     

Simple modules over generalized matrix rings

We study simple modules over the ring of generalized matrices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 245–247, 2007.     

A conjecture involving generalized matrix functions

This paper resolves the following conjecture of R. Merris: Let d^G_λ be the generalized matrix function determined by a subgroup G of the symmetric group S_m and an irreducible complex character λ of G. If A, B, and A?B are m-square positive semidefinite hermitian m-square matrices and d^G_λ(A)=d^G_λ(B)≠0, then A=B.