Matrices, Fermi Operators and Applications

TL;DR

This paper explores the mathematical properties of matrices, Fermi operators, and related operators, focusing on their algebraic structures, eigenvalues, and functions like exponentials, with applications to density and Kraus operators.

Contribution

It provides a comprehensive analysis of matrices and Fermi operators, including their commutators, eigenvalues, and matrix functions, contributing to the mathematical foundation of quantum operator theory.

Findings

Analysis of commutators and anticommutators of Fermi operators

Eigenvalue problem solutions for operators constructed from matrices and Fermi operators

Study of matrix functions such as exponentials in the context of these operators

Abstract

We consider the vector space of $n \times n$ matrices over $\mathbb C$, Fermi operators and operators constructed from these matrices and Fermi operators. The properties of these operators are studied with respect to the underlying matrices. The commutators, anticommutators, and the eigenvalue problem of such operators are also discussed. Other matrix functions such as the exponential functions are studied. Density operators and Kraus operators are also discussed.