matrix logarithm

If the matrix exponential climbs from the Lie algebra up to the group, the matrix logarithm climbs back DOWN: it takes a group element near the identity and recovers the algebra element (the generator) that produced it. It is the local inverse of exp, answering 'which infinitesimal symmetry did this finite one come from?'

Define log(A) by the power series log(A) = (A - I) - (A - I)^2/2 + (A - I)^3/3 - ..., which converges when A is close enough to I (precisely when ||A - I|| < 1). On that neighborhood log is a genuine inverse: exp(log A) = A and log(exp X) = X for small X. So exp and log together give a smooth chart identifying a patch of the group with a patch of its flat Lie algebra.

This is exactly why 'a Lie group near the identity looks like its Lie algebra' is literally true: log is the coordinate map that flattens the curved group onto the linear algebra. If A in SO(n) is near I, then log(A) is skew-symmetric; if A in SU(n), log(A) is anti-Hermitian with trace 0 — the logarithm lands back in the right Lie algebra.

Caveats are essential here. log is NOT globally defined: just as the scalar log is multivalued (e^{i theta} and e^{i(theta + 2 pi)} share a value), a rotation by 360 degrees equals the identity, so its log is ambiguous. Far from I, a real matrix may have no real logarithm at all (e.g. some matrices with negative eigenvalues), and the principal branch is the standard remedy.