Dirac Delta Function of Matrix Argument

TL;DR

This paper provides a pedagogical overview of the Dirac delta function extended to matrix arguments, including detailed proofs and potential applications in fields like Random Matrix Theory and Quantum Information.

Contribution

It systematically extends the Dirac delta function to vector and matrix spaces with detailed proofs, filling a gap in existing literature.

Findings

Extended Dirac delta to matrix arguments with proofs

Systematic discussion of delta function in vector and matrix spaces

Potential applications in Random Matrix Theory and Quantum Information

Abstract

Dirac delta function of matrix argument is employed frequently in the development of diverse fields such as Random Matrix Theory, Quantum Information Theory, etc. The purpose of the article is pedagogical, it begins by recalling detailed knowledge about Heaviside unit step function and Dirac delta function. Then its extensions of Dirac delta function to vector spaces and matrix spaces are discussed systematically, respectively. The detailed and elementary proofs of these results are provided. Though we have not seen these results formulated in the literature, there certainly are predecessors. Applications are also mentioned.