Research On N-D Polynomial Matrix Factorization

Polynomial matrix factorization plays an important role in engineeringcalculation such as symbolic computation and controlling, network coding, circuit,signal processing and multidimensional system. This thesis mainly reaches adecision that any polynomial matrix row-border theorem can obtain some goodnature about free module on the polynomial ring, and takes the lead research ofmatrix factorization on the Unique Factorization Domain （UFD）. The findingscontribute to improve and extend the relevant existing conferences and lay a goodfoundation for further study of the Matrix decomposition.This thesis is composed of five chapters. Chapter one covers a thoroughintroduction of this paper, which mainly narrates the historical background、 currentsituation of n-D polynomial matrix factorization and the main studies of it.Chapter two mainly introduces some related basic concepts and basic principlesin algebra used in this paper.Chapter three mainly discusses and obtains a theorem that any polynomialmatrix can be row-bordered up into a square polynomial matrix on the polynomialring. Illustrations are as follows: Any full row rank matrixC（z）∈A^l×m（l≤m）, it canbe incorporated into the first l row of a polynomial m×mmatrixC_e（z） whosedeterminant isφ_i（z）d, where φ_i（z）is independent of z_i.Chapter four, according to the row-border theorem obtains some good natureabout free module on the polynomial ring. This paper is mainly on the dissection offull row rank matrixF∈A^l×m（l≤m）, let d be the g.c.d. of all the l×lminorsof F, K=ρ（F）, we draw the following conclusions:（1）（K: d）/K=Torsion （A m/K）;（2） If K:d=K, then d=1;（3） Let f be a divisor of d, if K:f=K, then f=1;（4） K:f=Kiff f and d are relatively prime;（5） For any f∈A, let h=gcd（f, d）, then K: f=K:h.Chapter five mainly discusses the properties of matrix decomposition on UFD, and applies some conclusions of the matrix decomposition properties on thepolynomial ring to the unique factorization domain, and thus getting some newresults. One of the conclusions “Let A be UFD, for any full row rankmatrixF∈A^l×m（l≤m）,there exists a matrix V_i∈A^m×l such that FV_i=δ_iI_l” lays asolid foundation for the study of this chapter. Moreover, it also provides a goodtheoretical foundation for the further research of properties of matrix decompositionon unique factorization ring.