Generalizations of the Pfaffian to non-antisymmetric matrices
Mathematical Physics
2022-09-07 v1 math.MP
Abstract
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.
Keywords
Cite
@article{arxiv.2209.02578,
title = {Generalizations of the Pfaffian to non-antisymmetric matrices},
author = {Daniel Varjas},
journal= {arXiv preprint arXiv:2209.02578},
year = {2022}
}