math-riddle-may25.tex

math riddle of the day: Recall that orthogonal matrices are defined to be the (n by n) matrices over R^n for which (A^T A = A A^T = Identity), or equivalently, dotprod(x,y) = dotprod(Ax,Ay) for all x,y in R^n (inner-product preserving).
If A is an orthogonal matrix, we can show that its determinant must be either +-1.
Is the converse true? e.g. do all matrices with det 1 preserve inner products?
if so, prove it, if not, provide a counterexample.

i. Proof. First note that det A = det A^T by a handy theorem in Ch. 4 of FIS. Also det A det B = det AB by the same book.
So that A A^T = I implies (det A)^2 = 1 implies det A = +- 1.

ii. The converse is not true: let A be the matrix [1 1][1 0]. The basis vectors e_1, e_2 are sent to e_1, (1, 1) by this transformation
respectively. The dot product of the former is 0 (yay orthogonality) but the dot product of the latter is 1.

math riddle v2 (related to the number of fundamental flips). how many `fundamental' orientations of a space are there?
e.g. In R^2 there are two axis flips, but only one is necessary to get all rotations.

equivalently: can all det -1 rotations be obtained from a single flip and a det 1 rotation?

i. Yes (but I don't know how.)