$\begingroup$ The paper of Rosset and Rosset contains an explicit calculation for 2x2 matrices over a PID. The math review, and to some extent the paper, claim that it solves this for general square matrices over PIDs, but I do not see that this is the case. $\endgroup$ – Jack Schmidt Apr 18 '10 at 17:48

Let I be the 2 by 2 identity matrix. Then we prove that -I cannot be a commutator of two matrices with determinant 1. That is -I is not equal to ABA^{-1}B^{-1}. May 01, 2016 · If f has degree 4, we prove that all traceless matrices are contained in im f + im f, where f is over K and evaluated on M n (R) for n ≥ 3. This is not true for n = 2 ; a well-known counterexample is the polynomial f = [ x 1 , x 2 ] [ x 3 , x 4 ] + [ x 3 , x 4 ] [ x 1 , x 2 ] , which is central over M 2 ( K ) . where A = Mn(F), it is equal to either {0}, the set F1 of scalar matrices, the set [A,A] of traceless matrices, or A). This was then used in problems originally arising from Connes’ embedding conjecture and some other topics of functional analytic ﬂavor. Our goals in the present paper are diﬀerent, exist matrices A an.d B, both of which are P-orthogollal and P-skew-symmetric, sach that X = AB - BA. Methods for obtaining certain matrices which satisfy X = AB - BA are given. Methods are also given for determining pairs of anticommuting P-orth"gonal, P-skew-symmetric matrices. The last two lines state that the Pauli matrices anti-commute. The matrices are the Hermitian, Traceless matrices of dimension 2. Any 2 by 2 matrix can be written as a linear combination of the matrices and the identity. On commutators of matrices over unital rings Kaufman, Michael and Pasley, Lillian, Involve: A Journal of Mathematics, 2014; Identities for the zeros of entire functions of finite rank and spectral theory Anghel, N., Rocky Mountain Journal of Mathematics, 2019 with a traceless hermitian matrix L. It is conveniently expressed as a linear combination L = ~L ·~Λ ≡ NX2−1 j=1 L jΛ j, L j ∈ R, (8) with the set ~Λ forming a basis for traceless hermitian matrices, Λ j † = Λ j, called the generators. At the same time, they are a basis of the Lie algebra su(N) of SU(N), satisfying the commutation

## where A = Mn(F), it is equal to either {0}, the set F1 of scalar matrices, the set [A,A] of traceless matrices, or A). This was then used in problems originally arising from Connes’ embedding conjecture and some other topics of functional analytic ﬂavor. Our goals in the present paper are diﬀerent,

598 D Relations for Pauli and Dirac Matrices α iα j = 12 ⊗σ iσ j = σ iσ j 0 0 σ iσ j (D.7) so that commutators and anticommutators read α i,α j = 2i 3 ∑ k=1 ε ijkΣ k (D.8) ˆ α i,α j ˙ = 2δ ij14 and ˆ α i, β ˙ = 0 (D.9) The tensor product denoted by ‘⊗’ is to be evaluated according to the general prescription a11

### $\begingroup$ 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

Since these matrices represent physical variables, we expect them to be Hermitian. That is, they are equal to their conjugate transpose. Note that they are also traceless. As an example of the use of these matrices, let's compute an expectation value of in the matrix representation for the general state .