advanced32 min studyLesson 493 in Mathematics

Exponential maps and one-parameter subgroups

Integrate a constant identity generator into a global one-parameter motion while keeping the exponential's coordinate chart local.

Before you begin

On this page

The question for today

What you’ll understand

Construct the unique one-parameter subgroup with a prescribed identity velocity.
Explain local invertibility of exp and its possible global failures.
Compute a nontrivial triangular matrix exponential from its power series.

A tangent generator describes instantaneous motion; a one-parameter subgroup describes the same motion consistently for every elapsed parameter. The exponential map connects them. In a matrix group it agrees with the familiar convergent matrix power series, but the geometric construction works without choosing a matrix representation. Two facts must be kept separate: every generator gives a globally defined one-parameter subgroup, while the exponential map is guaranteed to be a coordinate chart only near zero. Global injectivity and global surjectivity require additional information about the group.

A complete invariant flow gives a subgroup

Start the left-invariant vector field associated with X at the identity and solve its ordinary differential equation. Local existence and uniqueness give an initial curve. Translation and the field's invariance extend it for every real time, and uniqueness yields the subgroup law: the value at s+t is the product of the values at s and t. Its velocity at zero is X. Conversely, differentiating any smooth one-parameter subgroup at zero recovers a generator, and the same uniqueness argument recovers the whole curve.

In matrices the curve solves gamma-prime equals gamma X with initial identity, so gamma(t)=exp(tX). The series converges for every finite matrix X, and the subgroup identity exp(sX)exp(tX)=exp((s+t)X) holds because the two generators are scalar multiples of the same matrix. It should not be extended to exp(X)exp(Y)=exp(X+Y) for unrelated noncommuting X and Y. The next lesson measures the missing commutator corrections.

\gamma_X(t)=\exp(tX),\qquad \gamma_X(s+t)=\gamma_X(s)\gamma_X(t),\qquad \gamma_X'(0)=X

Every identity tangent vector in a finite-dimensional Lie group determines one smooth one-parameter subgroup. In a matrix realization exp is the ordinary matrix exponential.↔ Scroll wide equations horizontally.

A local inverse does not remove global topology

The derivative of exp at zero is the identity on the Lie algebra. The inverse function theorem therefore makes exp a diffeomorphism between sufficiently small neighborhoods of zero and the group identity. A local logarithm is its inverse on that chosen image. The neighborhoods are part of the statement: beyond them, two different generators may exponentiate to the same group element, and a single logarithm may become multivalued or unavailable.

For the rotation generator J with J squared equal to minus identity, the exponential is cosine times identity plus sine times J. Its angle has period two pi, so global injectivity already fails for the circle group. Global surjectivity can also fail in other connected groups. For example, a real logarithm of diag(−2,−1/2) in SL₂(R) would commute with that diagonal matrix. Distinct eigenvalues force the logarithm to be real diagonal, whose exponential has positive diagonal entries, a contradiction.

\exp X=I+X+\frac{X^2}{2!}+\cdots,\qquad (d\exp)_0=\operatorname{id},\qquad \exp(\theta J)=\cos\theta\,I+\sin\theta\,J\quad(J^2=-I)

The series is the matrix formula. The identity derivative gives a local chart; the rotation formula explicitly shows why that chart is not globally one-to-one.↔ Scroll wide equations horizontally.

Functoriality makes the construction reusable

A smooth homomorphism carries one-parameter subgroups to one-parameter subgroups. Therefore applying the homomorphism after exp equals exponentiating its differential. This identity lets a representation transport Lie algebra calculations into matrices, or a covering map transport a motion between groups with different global topology. It does not imply that the differential uniquely determines a homomorphism from a disconnected source; it sees only the identity component unless extra information is supplied.

For numerical work, matrix interpolation from identity to a chosen endpoint is generally not its exponential trajectory. The straight segment can even leave a matrix subgroup, although an exact exponential of a tangent generator remains in that group. Distinguish a display interpolation, a numerical integration approximation, and the exact one-parameter law. Testing the subgroup identity on a calculated example is a useful check, especially when a scalar division in a closed-form formula has a removable zero-parameter singularity.

Work it through

Sum a triangular generator without assuming diagonalizability

For X=[[s,t],[0,0]], derive exp X. Evaluate s=log 2 and t=log 2, and identify the continuous s=0 limit.

For every integer k≥1, direct multiplication gives $X^k=\begin{pmatrix}s^k&s^{k-1}t\\0&0\end{pmatrix}$ . The top-right entry therefore collects the scalar exponential series with one power of s removed.
For s≠0, summing yields $\exp X=\begin{pmatrix}e^s&t(e^s-1)/s\\0&1\end{pmatrix}$ . At s=log 2 and t=log 2, the entries simplify to [[2,1],[0,1]], an affine-group element with positive scale.
As s tends to zero, (eˢ−1)/s tends to one. The formula therefore extends continuously to exp X=[[1,t],[0,1]]. In that case X²=0, so the power series independently gives I+X and confirms the removable limit.

The result

The exponential couples scale and translation; treating the two entries as independent scalar exponentials would miss the off-diagonal factor.

The living notebook

Explore the model

Interactive lab

Interactive lab · matrix

Inspect an endpoint without confusing its displayed path

Pause, Reset, then set a=2 and b=1 to inspect the exponential endpoint from the worked example. Predict determinant two. Repeat from Reset with both a=1 and b=1 for the nilpotent-generator endpoint. The instrument interpolates a unit square from identity to its selected matrix and back; that playback is not the exponential path exp(tX).

Interpolated geometryOriginal unit squareSelected final transform

Area scale · det A: 1.4
Singular values · σmax, σmin: 1.597, 0.876
Condition number · κ₂(A): 1.823

Diagonal scale a1.4Shear b0.6

t = 0 s

The selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAn exponential endpoint can be checked through matrix entries and determinant even when a display's interpolation uses a different path.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The identity derivative supplies a local inverse. Periodic rotations disprove global injectivity, the negative diagonal example disproves general surjectivity, and noncommuting generators require corrections to the addition rule.

Your turn

Try it yourself

Let E=E₁₂ in 2-by-2 matrices. Compute exp(4E), its square, and exp(8E), verifying the one-parameter subgroup identity directly.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

E²=0 makes every series term after the linear one vanish.

Compare with the solution

One has exp(4E)=I+4E=[[1,4],[0,1]]. Its square is I+8E because the product term 16E² vanishes. Also exp(8E)=I+8E=[[1,8],[0,1]], so the square equals exp(8E). This works for multiples of one fixed generator; it does not prove an additive exponential identity for two arbitrary noncommuting matrices.

Keep the idea

Take it with you

Every generator produces a unique global one-parameter subgroup.
The exponential has a guaranteed local inverse but can fail global injectivity and surjectivity.
A matrix endpoint, an exponential path, and an illustrative interpolation are different objects.

Read further

Original sources and deeper explanations to follow at your own pace.

Pavel Etingof: Lie Groups and Lie Algebras, exponential maps

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