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Adventures in 9-space

The three by three matrix is a workhorse for mechanicians, yet the study of this tool is seldom addressed mathematically. One reason is that the space of such matrices is nine-dimensional, quite beyond diagrams and  vision. The advantage of a mathematical investigation of the matrix space is clear in the study of two by two real matricies where rotations, shears, and squeezes correspond to area-preserving transformations of the plane. For transformations of space that preserve volume there is a much richer variety of phenomena that may correspond to mechanical events.

This blog will consider the 3 x 3 real matrices as a hypercomplex number system M(3,R) endowed with certain nilpotents. These nilpotents will be used to build the whole algebra from four three-dimensional subalgebras that arise from second order nilpotents. Let Z denote the zero matrix and Z1[(n,m)] denote the matrix with all zeroes except a one at row n and column m of the matrix. For instance, the identity is Z1[(1,1),(2,2),(3,3)].Let p = Z1[(1,2),(2,3)]. Check that p x p = Z1[(1,3)] and that p x p x p = Z. Then p is a nilpotent of the second order.

The nine-space will be profiled in terms of 3-dimensional subspaces like

S = { x + y p + z pp : x, y, z  in R }. An important curve is this subspace is

{exp(ap) = 1 + ap + (aa/2) pp : a in R}. This curve is a parabola in the plane of S given by x = 1. In the terminology of Sophus Lie it is a one-parameter subgroup of the multiplicative group of non-singular matrices. The curve is a conic section; the parabola in this case corresponds to the circle found in the complex plane, the unit hyperbola in the split-complex plane, or the line x = 1 in the dual number plane.

The subspace S of M(3,R) is spanned by p, pp, and the identity matrix. As this subspace is closed under products, it is a subalgebra. To more fully comprehend M(3,R) it is useful to consider p as one of four matrices that generate subalgebras isomorphic to S. 

Let q be the same as p except that the entry at (2,3) is changed to negative one.Let r = Z1[(2,1),(3,2)]. Finally, let s be the same as r except that the entry at (3,2) is changed to minus one. 

Readers at iMechanica should be able to confirm these matrix identities that arise from matrix multiplication: pp = – qq , rr = – ss, and pr = qs,   ps = qr,   rp = sq,   sp = rq. 

Another exercise that will confirm your engagement with M(3,R) will be the verification that 13 pr + 5 rp – ps – 7sp is twelve times the identity matrix. See if you can tell how the identity matrix can be obtained more generally. For instance, can you find a linear combination of three nilpotent products that gives the identity ?


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