I'll assume a square matrix with real entries in my answer. 1) A matrix with trace zero has both positive and negative eigenvalues, except if the matrix is the zero matrix. This is because the trace of a matrix is equal to the sum of its eigenva
There are several proofs of this nice result, differing in style and applicability. Some of them work for all ground fields, some for fields of characteristic [math]0[/math], and some only for [math]\R[/math] or [math]\C[/math]. I personally prefe 1. Rotations in 3D, so(3), and su(2). * version 2.0 Here [A,ˆ Bˆ] = AˆBˆ−BˆAˆ denotes the matrix commutator. Eq. (1.1.7) implies that the group of planar rotations is abelian: all possible group transformations commute. It means that the order in which a sequence of different rotations is applied to a vector in the plane does not matter: Rˆ(θ TRACELESS MATRICES THAT ARE NOT COMMUTATORS
There are several proofs of this nice result, differing in style and applicability. Some of them work for all ground fields, some for fields of characteristic [math]0[/math], and some only for [math]\R[/math] or [math]\C[/math]. I personally prefe
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.). The group operation is matrix multiplication.The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n
Lie algebra - encyclopedia article - Citizendium
Indeed, for any n ≥ 2, there exists an n × n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator Albert , Muckenhoupt : On matrices of trace zeros. 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; Characterization and Computation of Matrices of Maximal Trace Over Rotations Bernal, Javier and Lawrence, Jim, Journal of Geometry and An Example of a Matrix that Cannot Be a Commutator 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}. Commutation matrix - Wikipedia