A factorization of totally nonsingular matrices over a ring with identity

Random matrices over a DVR and LU factorization

LetR be a discrete valuation ring (DVR) andK be its fraction field. IfM is a matrix overR admitting a LU decomposition, it could happen that the entries of the factors L and U do not lie in R, but just inK. Having a good control on the valuations of these entries is very important for algorithmic applications. In the paper, we prove that in average these valuations are not too large and explain...

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n × n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is th...

A Theorem on Matrices over a Commutative Ring

Algebraic G-functions Associated to Matrices over a Group-ring

Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic G-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic n...

On the WZ Factorization of the Real and Integer Matrices

The textit{QIF}  (Quadrant Interlocking Factorization) method of Evans and Hatzopoulos solves linear equation systems using textit{WZ}  factorization. The  WZ factorization can be faster than the textit{LU} factorization  because,  it performs the simultaneous evaluation of two columns or two rows. Here, we present a  method for computing the real and integer textit{WZ} and  textit{ZW} factoriz...