基于Belitskii典范形的参数数的计算
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摘要
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.
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.
引文
[1] Sergeichuk V V.Canonical matrices for linear matrix problems.Linear Algebra Appl,2000,317:53–102
2 Drozd Y A.On tame and wild matrix problems.In:Matrix Problems(in Russian).Kiev:Inst Math Akad Nauk Ukrain SSR,1977,104–114
3 Kraft H,Riedtmann C.Geometry of representations of quivers.Representat Algebras,1986,116:109–145
4 de la Pena J A.On the dimension of the module-varieties of tame and wild algebras.Comm Algebra,1991,19:1795–1807
5 Xu Y G,Zhang Y B.Some geometrical properties of bocses(II).J Beijing Normal Univ(Nat Sci),2000,36:604–606
6 Zhang Y,Lei T,Bautista R.A matrix description of a wild category.Sci China Ser A,1998,41:461–475
7 Nagase H.τ-Wild algebras.In:Representations of Algebras(vol.II),Proceedings of the Ninth International Conference on Representations of Algebras.Beijing:Beijing Normal University Press,2000,365–372
8 Belitskii G R.Normal form in a space of matrices(in Russian).In:Analysis in Infinite-Dimensional Space and Operator Theory.Kiev:Naukova Dumka,1983,3–15
9 Belitskii G.Normal forms in matrix spaces.Integral Equations Operator Theory,2000,38:251–283
10 Gerstenhaber M.On nilalgebras and linear varieties of nilpotent matrices,III.Ann of Math(2),1959,70:167–205
11 Hesselink W.Singularities in the nilpotent scheme of a classical group.Trans Amer Math Soc,1976,222:1–32
12 Demmel J W,Edelman A.The dimension of matrices(matrix pencils)with given Jordan(Kronecker)canonical forms.Linear Algebra Appl,1995,230:61–87
13 Xu Y G,Zhang Y B.Indecomposability and the number of links.Sci China Ser A,2001,44:1515–1522
14 Arnol’d V I.On matrices depending on parameters.Usp Math Nauk,1971,26:101–114
15 Sergeichuk V V.A note on classification of holomorphis matrices up to similarity.Funct Anal Appl,1991,25:135–135
16 Friedland S.Analytic similarity of matrices.In:Algebraic and Geometric Methods in Linear Systems Theory.Lectures in Applied Mathematics,vol.18.Providence:Amer Math Soc,1981,43–85
17 Wu Y Q.Some canonical forms of a wild category.J Beijing Normal Univ(Nat Sci),2000,36:597–603
18 Kobal D.Belitskii’s canonical form for 5×5 upper triangular matrices under upper triangular similarity.Linear Algebra Appl,2005,403:178–182
19 Chen Y,Xu Y,Li H,et al.Belitskii’s canonical forms of upper triangular nilpotent matrices under upper triangular similarity.Linear Algebra Appl,2016,506:139–153
2 Drozd Y A.On tame and wild matrix problems.In:Matrix Problems(in Russian).Kiev:Inst Math Akad Nauk Ukrain SSR,1977,104–114
3 Kraft H,Riedtmann C.Geometry of representations of quivers.Representat Algebras,1986,116:109–145
4 de la Pena J A.On the dimension of the module-varieties of tame and wild algebras.Comm Algebra,1991,19:1795–1807
5 Xu Y G,Zhang Y B.Some geometrical properties of bocses(II).J Beijing Normal Univ(Nat Sci),2000,36:604–606
6 Zhang Y,Lei T,Bautista R.A matrix description of a wild category.Sci China Ser A,1998,41:461–475
7 Nagase H.τ-Wild algebras.In:Representations of Algebras(vol.II),Proceedings of the Ninth International Conference on Representations of Algebras.Beijing:Beijing Normal University Press,2000,365–372
8 Belitskii G R.Normal form in a space of matrices(in Russian).In:Analysis in Infinite-Dimensional Space and Operator Theory.Kiev:Naukova Dumka,1983,3–15
9 Belitskii G.Normal forms in matrix spaces.Integral Equations Operator Theory,2000,38:251–283
10 Gerstenhaber M.On nilalgebras and linear varieties of nilpotent matrices,III.Ann of Math(2),1959,70:167–205
11 Hesselink W.Singularities in the nilpotent scheme of a classical group.Trans Amer Math Soc,1976,222:1–32
12 Demmel J W,Edelman A.The dimension of matrices(matrix pencils)with given Jordan(Kronecker)canonical forms.Linear Algebra Appl,1995,230:61–87
13 Xu Y G,Zhang Y B.Indecomposability and the number of links.Sci China Ser A,2001,44:1515–1522
14 Arnol’d V I.On matrices depending on parameters.Usp Math Nauk,1971,26:101–114
15 Sergeichuk V V.A note on classification of holomorphis matrices up to similarity.Funct Anal Appl,1991,25:135–135
16 Friedland S.Analytic similarity of matrices.In:Algebraic and Geometric Methods in Linear Systems Theory.Lectures in Applied Mathematics,vol.18.Providence:Amer Math Soc,1981,43–85
17 Wu Y Q.Some canonical forms of a wild category.J Beijing Normal Univ(Nat Sci),2000,36:597–603
18 Kobal D.Belitskii’s canonical form for 5×5 upper triangular matrices under upper triangular similarity.Linear Algebra Appl,2005,403:178–182
19 Chen Y,Xu Y,Li H,et al.Belitskii’s canonical forms of upper triangular nilpotent matrices under upper triangular similarity.Linear Algebra Appl,2016,506:139–153