skew-symmetric matrices的正交相似类的Normal forms
Normal forms for orthogonal similarity classes of skew-symmetric matrices
摘要：
设F是特征不同于2的代数闭域。我们证明了F上的每个非奇异反对称n × n矩阵X与一个双对角反对称矩阵正交相似。在奇异情况下，还必须允许一些4对角块。若特征进一步为0，则构造了反对称矩阵的O_n（F）-相似类的标准形。在这种情况下，已知的正规形式（如在著名的书中所提出的由Mrumacher）是完全不同的。最后，我们研究了一些相关的矩阵。证明了n=2s+1的正规化幂零n × n双对角矩阵簇是s维不可约的.因此，反对称幂零双对角n × n矩阵被证明形成各种纯维数s。
英文摘要：
Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we construct the normal form for O_n(F)-similarity classes of skew-symmetric matrices. In this case the known normal forms (as presented in the well known book by Gantmacher) are quite different. Finally we study some related varieties of matrices. We prove that the variety of normalized nilpotent n by n bidiagonal matrices for n=2s+1 is irreducible of dimension s. As a consequence the skew-symmetric nilpotent bidiagonal n by n matrices are shown to form a variety of pure dimension s.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
