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Block Factorization of Hankel Matrices and Euclidean Algorithm

S. Belhaj (2010)

Mathematical Modelling of Natural Phenomena

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < ...

Clean matrices over commutative rings

Huanyin Chen (2009)

Czechoslovak Mathematical Journal

A matrix A ∈ M n ( R ) is e -clean provided there exists an idempotent E ∈ M n ( R ) such that A - E ∈ GL n ( R ) and det E = e . We get a general criterion of e -cleanness for the matrix [ [ a 1 , a 2 , ⋯ , a n + 1 ] ] . Under the n -stable range condition, it is shown that [ [ a 1 , a 2 , ⋯ , a n + 1 ] ] is 0 -clean iff ( a 1 , a 2 , ⋯ , a n + 1 ) = 1 . As an application, we prove that the 0 -cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n ≥ 3 . The analogous for ( s , 2 ) property is also obtained.

Combinatorial aspects of generalized complementary basic matrices

Miroslav Fiedler, Frank Hall (2013)

Open Mathematics

This paper extends some properties of the generalized complementary basic matrices, in particular, in a combinatorial direction. These include inheritance (such as for Alternating Sign Matrices), spectral, and sign pattern matrix (including sign nonsingularity) properties.

Conjugacy and factorization results on matrix groups

Thomas Laffey (1994)

Banach Center Publications

In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative...

Determinant Representations of Sequences: A Survey

A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy (2014)

Special Matrices

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this...

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