Representation formulae for the determinant in a neighborhood of the identity

TL;DR

This paper derives integral and power series representations for the inverse determinant function near the identity matrix, linking these to coordinate transformations of the Dirac delta distribution.

Contribution

It introduces new integral and power series formulas for the inverse determinant function around the identity, connecting to distribution change-of-variables.

Findings

Provides explicit integral representation of det(A)^{-1} near identity.

Develops power series expansion for det(A)^{-1} in a neighborhood of identity.

Establishes a connection between determinant formulas and Dirac delta distribution transformations.

Abstract

We prove an integral representation and a power series expansion for the function $\det(A)^{-1}$ in a small neighborhood of the identity matrix. Both results are closely linked to the formula for the change of coordinates of the Dirac delta distribution in $\mathbb{R}^m$.