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 $p={\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}}$. Using matrix multiplication, the reader will compute matrices p² and p³. Next, consider

$q={\begin{pmatrix}0&1&0\\0&0&-1\\0&0&0\end{pmatrix}}\quad r={\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}}\quad s={\begin{pmatrix}0&0&0\\1&0&0\\0&-1&0\end{pmatrix}}$.

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 $T_{p}=\{xI+yp+zpp:x,y,z\in R\}$. In fact, T_p 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:

$\exp(tp)=1+tp+{\frac {t^{2}}{2}}p^{2}.$ The parameter t traces out a parabola in the plane x = 1 of T_p.

The parabola is in fact a group isomorphic to (R, +) since $\exp(tp)\times \exp(sp)=\exp((s+t)p)$. In contrast to orthogonal subgroups of GL(3,R), this group is not compact.

Exercise : Show there are similar one-parameter subgroups in subspaces $T_{s},\ T_{q},\ T_{r}$ of M(3,R).