[Paper Review] Trace evaluation of matrix determinants and inversion of 4 $ imes$ 4 matrices in terms of Dirac covariants

Frieder Kleefeld, M. Dillig|ArXiv.org|Jun 19, 1998

Matrix Theory and Algorithms3 citations

TL;DR

This paper presents a novel method for computing determinants and inverses of 4×4 matrices using Dirac covariants, leveraging trace identities and the algebraic structure of Dirac matrices. The key contribution is a closed-form expression for the inverse of a 4×4 matrix in terms of its Dirac covariant components, enabling efficient computation in high-energy physics applications such as spinor and gamma matrix calculations.

ABSTRACT

In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d imes d$ matrices, if their traces are known. As a next step $4 imes 4$ matrices are expressed in terms of Dirac covariants. The third step is the calculation of the corresponding inverse matrices in terms of Dirac covariants.

Motivation & Objective

• To derive a general method for evaluating determinants of d×d matrices using only trace information.
• To express arbitrary 4×4 matrices in terms of Dirac covariants, simplifying their algebraic treatment.
• To provide a closed-form expression for the inverse of a 4×4 matrix in terms of its Dirac covariant components.
• To facilitate efficient computation in quantum field theory and phenomenology, particularly in spinor and gamma matrix formalisms.

Proposed method

• The paper uses trace identities to express the determinant of a d×d matrix as a function of its traces, enabling determinant evaluation without full matrix decomposition.
• It decomposes a 4×4 matrix into a linear combination of Dirac gamma matrices and the identity, forming a basis of 16 independent Dirac covariants.
• The inverse of the matrix is constructed by expressing it as a linear combination of the same Dirac covariants, using trace orthogonality and projection techniques.
• The method relies on the algebraic properties of the Dirac gamma matrices, including their anticommutation relations and trace identities.
• The inverse is derived via a systematic projection onto the Dirac basis, ensuring consistency with matrix inversion axioms.
• The approach is validated through explicit examples and consistency checks in the context of high-energy physics.

Experimental results

Research questions

• RQ1 How can the determinant of a d×d matrix be computed solely from its trace information?
• RQ2 What is the most compact and efficient representation of a 4×4 matrix using Dirac covariants?
• RQ3 How can the inverse of a 4×4 matrix be expressed in terms of its Dirac covariant components?
• RQ4 What algebraic identities enable trace-based evaluation of matrix invariants in 4×4 matrix theory?
• RQ5 Can the inverse of a 4×4 matrix be systematically derived using the Dirac basis without explicit matrix inversion?

Key findings

• The determinant of a d×d matrix can be evaluated using only trace information, provided the traces of all powers up to d are known.
• Any 4×4 matrix can be uniquely expressed as a linear combination of the 16 Dirac covariants: I, γμ, σμν, γ5γμ, γ5.
• The inverse of a 4×4 matrix is explicitly constructed as a linear combination of the same 16 Dirac covariants, with coefficients derived from traces of the original matrix and its products.
• The method enables efficient computation of matrix invariants and inverses in quantum field theory, particularly in spinor and gamma matrix calculations.
• The formalism is consistent and closed under matrix operations, with all results expressible in terms of trace evaluations.
• The approach significantly reduces computational complexity compared to standard matrix inversion, especially in high-energy physics contexts.