基于Belitskii典范形的参数数的计算

Calculation of the number of parameters via Belitskii's canonical forms

设(Λ,M)是一个线性矩阵问题,基于Belitskii典范形,本文首先得到了(Λ,M)上的任一矩阵共轭轨道维数及余维数的计算公式,并由此建立了参数数与不可分解典范形的参数簇之间的关系,即证明参数数μ(ind M_n)等于每个不可分解参数化典范形中参数数目的极大值,从而提供了基于Belitskii典范形计算(Λ,M)上的矩阵的共轭轨道维数与参数数μ(ind M_n)的有效方法.作为应用,本文计算了Wascow问题、矩阵束问题及上三角相似变换下的幂零上三角矩阵问题的参数数.

For a linear matrix problem（Λ, M）, we first obtain the formulae of dimension and codimension of conjugacy orbit of any matrix over（Λ, M）, and thus establish the relationship between parametric families of indecomposable canonical forms and the number of parameters μ（ind M） defined in terms of algebraic geometry and transformation group. More precisely, we show that μ（ind M_n） is just the maximum of the number of parameters contained in each parametric indecomposable canonical matrix in M_n, which provides an effective way to calculate dimensions of conjugacy orbits of matrices over（Λ, M） and the number of parameters μ（ind M_n）based on Belitskii＇s algorithm. As applications, we calculate the number of parameters of Wascow＇s canonical forms, matrix pencils and upper triangular nilpotent matrices under upper triangular similarity.