Generalizations of the Pfaffian to non-antisymmetric matrices

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 squ...

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Bibliographic Details
Published inarXiv.org
Main Author Varjas, Daniel
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.09.2022
Subjects
Polynomials
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Summary: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.
ISSN:2331-8422