Abstract Algebra/3x3 real matrices

The algebra M(3,R) of 3 x 3 real matrices, having nine dimensions, is well beyond visual scope. Nevertheless, study of nilpotents of second degree brings to light some shiny facets. Alternative basis Let . Using matrix multiplication, the reader will compute matrices p2 and p3. Next, consider . Exercises: 1) Show p p = − q q. 2) Show r r = − s s. 3) Show pr = qs. 4) Show p s = q r. 5) Show r p = s q. 6) Show s p = r q. 7) Show pr + ps + 2sp is twice the identity matrix. 8) Show that { pr, ps, rp, sp } is a linearly dependent subset of M(3,R). 9) Show that { p, q, r, s, pp, rr, pr, ps, rp } is a basis for M(3,R). One-parameter subgroups of GL(3,R) Consider the subspace . In fact, Tp is a 3-dimensional subalgebra of M(3,R). Since p is nilpotent of second degree, the exponential series for tp , t in R, has only three terms: The parameter t traces out a parabola in the plane x = 1 of Tp. The parabola is in fact a group isomorphic to (R, +) since . In contrast to orthogonal subgroups of GL(3,R), this group is not compact. Exercise : Show there are similar one-parameter subgroups in subspaces of M(3,R).