Speaking of infinite matrix products -- I recently had occasion to re-invent product integrals of matrices (or on Lie groups), which turn out to have been originally invented by Volterra, as far as I can tell: Prodint_{a <= t <= b} M(t)^dt is defined as lim as N -> oo of (M(t_N)^del_t) * . . . * (M(t_1)^del_t) where M: [a,b] -> Iso[R^p] is (say) a piecewise smooth map into p x p real matrices, {a = t_0 < . . . < t_N = b} is a partition P_N of [a,b] with mesh(P_N) -> 0, and Iso(R^p] is all invertible p x p real matrices. Alternatively, the given function can be A: [a,b} -> End[R^p], where End[R^p] is all real p x p matrices, and Prodint_{a <= t <= b} exp(A(t) dt) is defined as lim as N -> oo of exp(A(t_N) del_t) * . . . * exp(A(t_1) del_t) blah blah blah. Man, these are hard to evaluate even in simple cases!!! --Dan On Aug 28, 2014, at 5:10 AM, Bill Gosper <billgosper@gmail.com> wrote:
Download gosper.org/FDrags128.mp4 . Then nth Fourier coefficient is a bit hairy--infinite product of nontriangular 3x3 matrices. --rwg (I probably should have exported a .gif.) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun