matrix exponential

The matrix exponential takes the familiar power series for e^x and feeds a matrix into it. The result turns a Lie algebra element (a 'velocity', a direction at the identity) into an actual group element (a finite transformation). It is the engine that converts infinitesimal symmetry into finite symmetry.

Define exp(A) = sum_{k>=0} A^k / k! = I + A + A^2/2 + A^3/6 + .... This series converges for EVERY square matrix A (the norms are bounded by the scalar series for e^||A||), so exp is defined on all of M_n. Key identities: exp(0) = I, d/dt exp(tA) = A exp(tA), and exp(A)^-1 = exp(-A). But beware: exp(A+B) = exp(A)exp(B) holds ONLY when A and B commute.

Its starring role here: exp maps a Lie algebra g into its Lie group G. If A is skew-symmetric then exp(A) is orthogonal; if A is anti-Hermitian then exp(A) is unitary; if trace A = 0 then det exp(A) = 1, so exp sends sl(n) into SL(n). The identity det(exp A) = exp(trace A) is the determinant-trace bridge that makes all of this work.

It is also how you SOLVE linear ODEs: the system x'(t) = A x(t) with x(0) = x0 has the exact solution x(t) = exp(tA) x0. So the matrix exponential is simultaneously the flow of a linear dynamical system and the one-parameter subgroup t -> exp(tA). A caveat: to compute it, diagonalize or use the Jordan form — naively summing the series is slow and unstable.