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arXiv · 2209.02578

Generalizations of the Pfaffian to non-antisymmetric matrices

Daniel Varjas
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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.

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Daniel Varjas. 2022-09-06. Generalizations of the Pfaffian to non-antisymmetric matrices. https://arxiv.org/abs/2209.02578

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