Generalized commutators in matrix rings: Linear and

Key words: Commutator; matrix; orthogonal; skew-symmetric. 1. Introduction Let P = Ip -+ ( -I q), the direct sum of the p X P identity matrix and the negative of the q X q identity matrix. Katz and Olkin [IF define a matrix A to be orthogonal with respect to P (p-orthog onal) if only if APA'=P (1) where A' is the transpose of the matrix A. So I've had a read of this, and I'm still not convinced as to why gauge fields are traceless and Hermitian.I follow the article fine, it's just the section that says "don't worry about this complicated maths, the point is that the gauge field is in the Lie algebra". (a3) they must be traceless (the trace of a square matrix is the sum of its diagonal elements). This results from the commutation relations (A-01,02,03) and the property that the trace of the product of two square matrices is independent of their order : \begin{equation} C=[A,B]=AB-BA \Longrightarrow TrC=Tr[A,B]=Tr(AB)-Tr(BA)=0 \tag{A-08} \end The action of a rotation R(θ) can be represented as 2×2 matrix: x y → x′ y′ = cosθ −sinθ sinθ cosθ x y (4.2) Exercise 4.1.1 Check the formula above, then repeat it until you are sure you know it by heart!! Intuitively two successive rotations by θand ψyield a rotation by θ+ ψ, and hence the group of A is a scalar matrix, we may assume that X is a diagonal matrix. However, if X is a diagonal matrix, then XY — YX has all diagonal entries equal to zero. Since this contradicts A = al where a / 0, the theorem follows. Added in proof. Theorem 1 of a paper by Joel Anderson and Joe Parker, From now on we consider only vacuum solutions. Suppose we use the Lorenz gauge. As shown above, we still have the freedom of coordinate transformations with $\square \xi^\mu =0$, which preserve the gauge. A quantitative version of the commutator theorem for zero trace matrices William B. Johnsony, Narutaka Ozawa z, Gideon Schechtman x Abstract Let A be a m m complex matrix with zero trace and let " > 0. Then there are m m matrices B and C such that A = [B;C] and kBkkCk K "m"kAkwhere K " depends only on ". Moreover, the matrix B can be taken to

## In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.This is the normal subgroup of the general linear group given by the kernel of the determinant: (,) → ×. where we write F × for the multiplicative group of F (that is, F excluding 0).

Covariant Formulation of Electrodynamics is the continuity equation. Note that (as Jackson remarks) this only works because electric charge is a Lorentz invariant and so is a four-dimensional volume element (since ). Next, consider the wave equations for the potentials in the Lorentz gauge (note well that Jackson for no obvious reason I can see still uses Gaussian units in this part of chapter 11, which is goiing to make this a pain

### Explicitly, in the large N limit, the equation of motion for Ao provides the non-commutativity of the coordinates, which leads to the classical matrix commutator (9). Such a regularised matrix model associated with the spinor matrix field z can be obtained by following the construction of Polychronakos for the finite matrix model on the plane [17].

$\begingroup$ I see, but wouldn't this show that traceless matrices are sums of commutators? I want to know wether any traceless matrix is a commutator. I'll edit my question to make it clearer. $\endgroup$ – Olivier Bégassat Mar 27 '12 at 21:57 linear algebra - Is there a way to find $B,C$ such that $A $\begingroup$ Can you explain why $\mathfrak{sl}_2(\mathbb{C})$ being semisimple implies that every matrix can be expressed as a commutator? I can see that implies it can be written as a sum of commutators, but not why it should be a single one. $\endgroup$ – Nate Mar 28 '14 at 4:02