Defining exp of a matrix
The scalar identity e^x = 1 + x + x^2/2! + x^3/3! + ... uses only addition, multiplication, and powers — all of which matrices have. So define the matrix exponential exp(A) = I + A + A^2/2! + A^3/3! + ... . This series converges for every square matrix A, so exp(A) is always defined. It is the single most important map in Lie theory.
A worked rotation
The cleanest example: exponentiate the antisymmetric matrix J = [0, -1; 1, 0]. Powers cycle: J^2 = -I, J^3 = -J, J^4 = I. Grouping the series by even and odd powers reproduces the cosine and sine series exactly, giving a rotation by angle t. A flat, simple generator becomes a curved member of SO(2).
J = [0, -1; 1, 0]
exp(t*J) = I + tJ + (t^2/2!)J^2 + (t^3/3!)J^3 + ...
= (1 - t^2/2! + ...) I + (t - t^3/3! + ...) J
= cos(t) I + sin(t) J
= [cos t, -sin t; sin t, cos t] = R(t) <- a rotation!
# check: exp(t*J) is in SO(2), and t controls the angle smoothly.
# det( exp(tJ) ) = e^{ trace(tJ) } = e^{0} = 1 (always)The two laws that make it a bridge
Two facts make exp a bridge from a flat space to a curved group. First, det(exp A) = e^{trace(A)}, so if trace(A) = 0 then det(exp A) = 1 and exp A lands in SL(n). Second, exp(A) is always invertible with inverse exp(-A). So traceless matrices (a simple linear condition) map onto SL(n) (a curved group). The flat domain is a Lie algebra; the curved target is a Lie group.
The return trip: matrix logarithm
The matrix logarithm inverts exp near the identity: log(I + X) = X - X^2/2 + X^3/3 - ... . For a group element g close to I, log(g) lands back in the Lie algebra, recovering the generator. exp and log together let you translate freely between the curved group and its flat tangent space at the identity — and the flat space is far easier to compute in.
Letting the parameter t run continuously, t |-> exp(tA) traces a one-parameter subgroup — a smooth curve through the identity that respects the group law. The next guide takes this curve as the heart of the theory.