Pfaffians and Skew‐Symmetric Matrices
Pfaffians and Skew‐Symmetric Matrices
复制标题
DOI：
10.1112/plms/s3-19.4.730
复制
发表时间：
1969-10
影响因子：
1.8
通讯作者：
P. Heymans
中科院分区：
文献类型：
--
作者：
P. Heymans
本文主要讨论了三个问题。第一种是将斜对称矩阵的子式展开为pfaffian中的二次型。这可以通过多种方式来实现，我将详细讨论一个特定的展开，我称之为规范展开。第二个问题是一种相反的情况：我证明了一对Pfaffian的乘积可以扩展为未成年人的线性组合。由此推论，Pfaffian中的每个二次型要么恒为零(被认为是矩阵元素中的多项式)，要么可以表示为子式中的线性形式。第三个问题是在pfaffian中求那些相同为零的二次型的线性系统的基。证明了与(k+1)x(jc+1)反对称矩阵相关的这种线性无关零型的个数是的，并给出了它们的显式表达式。关于斜对称矩阵及其与Pfaffian的关系的文献非常广泛，可以追溯到Cayley(10)，他证明了奇数阶斜对称行列式为零，偶数阶斜对称行列式是Pfaffian的平方；他还证明了与主对角线相交有r-1个零点的斜对称矩阵的r×r次项等于两个Pfaffian的乘积。在接下来的50年里，这些结果和其他与之密切相关的结果被用不同的方法再次证明了六次以上。1903年，布里尔(Brill)的一篇论文首次将一般小调展开为pfaffian中的二次形式。我证明了他的展开式是一类类似的展开式，并给出了一个较短的证明。我论文的第二部分似乎没有发表任何结果，第三部分只有几个孤立的结果。(见(1)、(2)、(4))。
There are three main problems discussed in this paper. The first is the expansion of a minor of a skew-symmetric matrix as a quadratic form in pfaffians. This can be done in many ways and I discuss in detail one particular expansion, that I call canonical. The second problem is a sort of converse: I prove that a product of a pair of pfaffians may be expanded as a linear combination of minors. It follows that every quadratic form in pfaffians either is identically zero (considered as a polynomial in the elements of the matrix) or is representable as a linear form in the minors. The third problem is the obtaining of a basis for the linear system of those quadratic forms in pfaffians which are identically zero. I show that the number of such linearly independent zero-forms, associated with a (k+ 1) x (Jc+ 1) skew-symmetric matrix, is and I give explicit expressions for these. The literature on skew-symmetric matrices and their connexion with pfaffians is very extensive and goes back to Cayley (10), who proved that a skew-symmetric determinant of odd order is zero and that one of even order is the square of a pfaffian; he also showed that an r x r minor of a skew-symmetric matrix, which has r—1 zeros by intersection with the leading diagonal, is equal to the product of two pfaffians. These results and others closely related to them were proved again by different methods more than half a dozen times in the course of the next fifty years. The only published expansion of a general minor as a quadratic form in the pfaffians appeared for the first time in 1903 in a paper by Brill (2). I exhibit his expansion as one of a general class of similar expansions and give a shorter proof. No results appear to have been published on the second section of my paper and only a few isolated ones on the third section.(See (1),(2),(4).)