On the factorization of polynomial matrices over the domain of principal ideals

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.     

Intermediate subgroups of chevalley groups over a field of fractions of a ring of principal ideals

Let K be a field of fractions of a principal ideal ring R and G_K be a Chevalley group (of normal type) over K. For each subring P ⊂ K, denote by G_P a subgroup of all elements of G_K with coefficients in P. Let M be intermediate between G_R and G_K, i.e., G_R ⊆ M ⊆ G_K. We prove that M=G_P for some intermediate subring P (R ⊆ P ⊆ K). Supported by RFFR grant No. 96-01-00409. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 347–358, May–June, 2000.     

On annihilator ideals of a polynomial ring over a noncommutative ring

Let R be a ring and let R[x] denote the polynomial ring over R. We study relations between the set of annihilators in R and the set of annihilators in R[x].     

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.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1581–1584, November, 1995.     

A criterion for the splitting of a principal bundle over a projective space

Let E_G be an algebraic principal G-bundle over \mathbbC\mathbbPⁿ ,\mathbb{C}\mathbb{P}^n , n  \mathbbC.\mathbb{C}. We prove that E_G admits a reduction of structure group to a one-parameter subgroup of G if and only if

Factorization of transformations over a local ring

Let V be a finitely generated free module over a local ring R, and π an invertible linear transformation for V. Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R. The smallest number of factors needed is not greater than r+1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.     

Sections of zero dimensional ideals over a Noetherian ring

Let A be a commutative Noetherian ring and be an ideal containing a monic polynomial such that A[T]/I is zero dimensional. Suppose the conormal module I/I ² is generated by r elements over A[T]/I. Then a set of r generators of can be lifted to a r generating set of I. A part of this work is done at the Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Received: 12 March 2006 Revised: 29 January 2007     

Scaling of matrices to achieve specified row and column sums

IfA is ann ×n matrix with strictly positive elements, then according to a theorem ofSinkhorn, there exist diagonal matricesD ₁ andD ₂ with strictly positive diagonal elements such thatD ₁ A D ₂ is doubly stochastic. This note offers an alternative proof of a generalization due toBrualdi, Parter andScheider, and independently toSinkhorn andKnopp, who show that A need not be strictly positive, but only fully indecomposable. In addition, we show that the same scaling is possible (withD ₁ =D ₂) whenA is strictly copositive, and also discuss related scaling for rectangular matrices. The proofs given show thatD ₁ andD ₂ can be obtained as the solution of an appropriate extremal problem.The scaled matrixD ₁ A D ₂ is of interest in connection with the problem of estimating the transition matrix of a Markov chain which is known to be doubly stochastic. The scaling may also be of interest as an aid in numerical computations.Research sponsored in part by the Boeing Scientific Research Laboratories.     

A criterion for the existence of common invariant subspaces of matrices

It is shown that square matrices $\text{A}$ and $\text{B}$ have a common invariant subspace W of dimension $\text{k⩾1}$ if and only if for some scalar s, $\text{A+sI}$ and $\text{B+sI}$ are invertible and their kth compounds have a common eigenvector, which is a Grassmann representative for $\text{W}$. The applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated.     

Generalized doubly stochastic and permutation matrices over a ring

Let R be a ring with unity. A combinatorial argument is used to show that the R-module Δ_n(R) of all n × n matrices over R with constant row and column sums has a basis consisting of permutation matrices. This is used to characterize orthogonal matrices which are linear combinations of permutation matrices. It is shown that all bases of Δ_n(R) consisting of permutation matrices have the same cardinality, and other properties of bases of Δ_n(R) are investigated.     

A criterion for the strongly semistable principal bundles over a curve in positive characteristic

Let X be an irreducible smooth projective curve over an algebraically closed field k of positive characteristic and G a simple linear algebraic group over k. Fix a proper parabolic subgroup P of G and a nontrivial anti-dominant character λ of P. Given a principal G-bundle E_G over X, let E_G(λ) be the line bundle over E_G/P associated to the principal P-bundle E_G→E_G/P for the character λ. We prove that E_G is strongly semistable if and only if the line bundle E_G(λ) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles.     

Minimum positive determinant of integer matrices with constant row and column sums

The least possible positive determinant of zero-one matrices that have constant row and column sums is determined, thus proving a conjecture of Newman. The result is extended to n×n integer matrices.