论文标题
Pfaffian对非抗对称矩阵的概括
Generalizations of the Pfaffian to non-antisymmetric matrices
论文作者
论文摘要
我们研究了Pfaffian对非抗对称矩阵的两个概括，并彼此得出了它们的性质和关系。第一种方法是基于Wigner正常形式的，适用于偶联正常矩阵，并保留了Pfaffian的大多数特性，包括它是决定符的正方形。第二种方法是采用适用于所有矩阵的反对称矩阵的pfaffian。我们表明，该公式等同于将非抗对称基质取代为Pfaffian的多项式定义。我们发现，两个定义在一个积极的实际因素上有所不同，这使得第二个定义违反了决定性身份。
We study two generalizations of the Pfaffian to non-antisymmetric matrices and derive their properties and relation to each other. The first approach is based on the Wigner normal-form, applicable to conjugate-normal matrices, and retains most properties of the Pfaffian, including that it is the square-root of the determinant. The second approach is to take the Pfaffian of the antisymmetrized matrix, applicable to all matrices. We show that this formulation is equivalent to substituting a non-antisymmetric matrix into the polynomial definition of the Pfaffian. We find that the two definitions differ in a positive real factor, making the second definition violate the determinant identity.
