Matrix Exponential and One-Parameter Flows

Derivation

Matrix Exponential and One-Parameter Flows

D-088 Home PU-104 Threads symmetry · waves Depends on Spectral Theorem for Self-Adjoint Operators, Unitary Operators as Inner-Product Isometries

Statement

Let \(A\) be a bounded linear operator on a finite-dimensional (or, more generally, a Hilbert) space \(V\). The operator series \(\exp(tA)=\sum_{n=0}^{\infty}\frac{t^{n}A^{n}}{n!}\) converges in operator norm for every \(t\in\mathbb{R}\) and defines the unique differentiable operator-valued function \(U(t)\) satisfying \(\dot U=AU\), \(U(0)=I\). Consequently \(x(t)=U(t)x_{0}\) is the unique solution of the initial-value problem \(\dot x=Ax\), \(x(0)=x_{0}\), the family \(\{U(t)\}_{t\in\mathbb{R}}\) is a one-parameter group, and \(U(t)\) is unitary for all \(t\) if and only if the generator is skew-adjoint, \(A^{\dagger}=-A\) — equivalently \(A=-\tfrac{i}{\hbar}H\) with \(H\) self-adjoint, in which case \(H\) is the physical generator of the flow.

Why it matters

Every linear, time-translation-invariant evolution in physics — free rigid-body rotation, small oscillations about equilibrium, RLC transients, and above all Schrödinger dynamics — is a one-parameter flow. The matrix exponential is the single object that packages "run the equations of motion forward by a time \(t\)" into an operator, and it turns the differential law \(\dot x=Ax\) into the algebraic law \(x(t)=e^{tA}x_{0}\).

The self-adjoint case is the mathematical heart of quantum mechanics: Stone's theorem says a strongly continuous one-parameter unitary group is exactly \(e^{-iHt/\hbar}\) for a self-adjoint \(H\). Conservation of probability, reversibility, and the spectral interpretation of energy all descend from the single fact that a self-adjoint generator produces a unitary flow.

Assumptions

The generator \(A\) is time-independent.If \(A=A(t)\) the solution is no longer \(e^{tA}\); one needs the time-ordered exponential \(U(t)=\mathcal{T}\exp\!\int_{0}^{t}A(s)\,ds\), and the simple group law \(U(s+t)=U(s)U(t)\) fails.

\(A\) is bounded (automatic in finite dimensions).For an unbounded self-adjoint \(H\) on infinite-dimensional \(V\), the power series need not converge; the exponential must be built from the spectral resolution (Stone's theorem), and \(H\) acts only on a dense domain.

The inner product defining "self-adjoint" is the physical one.Unitarity is relative to a metric. If the true conserved bilinear form is indefinite (e.g. a \(\mathcal{PT}\)-symmetric or Krein-space problem), \(A^{\dagger}=-A\) in the wrong metric gives a flow that is not norm-preserving and eigenvalues may leave the imaginary axis.

The space is complete (a Banach/Hilbert space).Norm convergence of the partial sums to a limit operator requires completeness; without it the series defines a Cauchy sequence with no limit and \(\exp(tA)\) is not well defined.

Derivation

\[ \left\lVert \sum_{n=N}^{M}\frac{t^{n}A^{n}}{n!}\right\rVert \le \sum_{n=N}^{M}\frac{|t|^{n}\,\lVert A\rVert^{n}}{n!}\;\xrightarrow[N\to\infty]{}\;0 \]

Submultiplicativity \(\lVert A^{n}\rVert\le\lVert A\rVert^{n}\) plus the tail of the scalar series for \(e^{|t|\,\lVert A\rVert}\); the partial sums are Cauchy in operator norm, and completeness gives a limit \(\exp(tA)\). B

\[ \frac{d}{dt}\exp(tA)=\sum_{n=1}^{\infty}\frac{n\,t^{n-1}A^{n}}{n!}=A\sum_{n=1}^{\infty}\frac{t^{n-1}A^{n-1}}{(n-1)!}=A\,\exp(tA)=\exp(tA)\,A \]

The series converges uniformly on compact \(t\)-intervals (step 1 bound), so term-by-term differentiation is legal; factoring \(A\) out on either side is allowed because \(A\) commutes with its own powers. This establishes \(\dot U=AU\), \(U(0)=I\). C

\[ \exp(sA)\exp(tA)=\sum_{j,k\ge0}\frac{s^{j}t^{k}}{j!\,k!}A^{j+k}=\sum_{n\ge0}\frac{A^{n}}{n!}\sum_{m=0}^{n}\binom{n}{m}s^{m}t^{n-m}=\exp\!\big((s+t)A\big) \]

Cauchy product of two absolutely convergent series, then the binomial theorem on the commuting scalars \(s,t\); this is the one-parameter group law. In particular \(\exp(tA)\exp(-tA)=I\), so every \(U(t)\) is invertible with \(U(t)^{-1}=U(-t)\). B

\[ x(t):=\exp(tA)\,x_{0}\;\Rightarrow\;\dot x(t)=A\exp(tA)x_{0}=Ax(t),\qquad x(0)=x_{0} \]

Apply step 2 to the constant vector \(x_{0}\); the exponential thus solves the initial-value problem. A

\[ y(t)\ \text{any solution};\quad w(t):=\exp(-tA)\,y(t),\quad \dot w=-A\,e^{-tA}y+e^{-tA}\dot y=e^{-tA}(\dot y-Ay)=0 \]

Product rule with step 2 and \(\dot y=Ay\); \(w\) is constant, so \(w(t)=w(0)=y_{0}\), giving \(y(t)=\exp(tA)y_{0}\). Two solutions with the same initial data coincide — uniqueness of the flow. C

\[ \frac{d}{dt}\big(U(t)^{\dagger}U(t)\big)=\dot U^{\dagger}U+U^{\dagger}\dot U=U^{\dagger}\big(A^{\dagger}+A\big)U \]

Adjoint of \(\dot U=AU\) is \(\dot U^{\dagger}=U^{\dagger}A^{\dagger}\); combine and factor. The bracket vanishes precisely when \(A^{\dagger}=-A\). C

\[ A^{\dagger}=-A\ \Rightarrow\ \frac{d}{dt}\big(U^{\dagger}U\big)=0,\quad U(0)^{\dagger}U(0)=I\ \Rightarrow\ U(t)^{\dagger}U(t)=I\ \ \forall t \]

A vanishing derivative with initial value \(I\) forces \(U^{\dagger}U\equiv I\); on a finite-dimensional space this is unitarity. Writing \(A=-\tfrac{i}{\hbar}H\), skew-adjointness of \(A\) is exactly self-adjointness of \(H\). C

\[ H=\sum_{k}\lambda_{k}P_{k}\ \Rightarrow\ U(t)=\exp\!\Big(-\tfrac{i}{\hbar}tH\Big)=\sum_{k}e^{-i\lambda_{k}t/\hbar}\,P_{k},\qquad \big|e^{-i\lambda_{k}t/\hbar}\big|=1 \]

By the spectral theorem for self-adjoint operators \(H\) has a real orthogonal eigenprojection resolution; the exponential acts eigenspace-by-eigenspace, multiplying each by a unit-modulus phase. Preservation of the inner product then follows because unitary operators preserve it. C

Result

\[ x(t)=\exp(tA)\,x_{0},\qquad \frac{d}{dt}\exp(tA)=A\exp(tA),\qquad A^{\dagger}=-A\iff U(t)\ \text{unitary} \]

Reading. The matrix exponential is the propagator: it advances any initial state by a time \(t\) along the linear flow \(\dot x=Ax\), and it is the only operator that does so. Composing two advances adds their times (the group law), reversing time inverts the operator, and when the generator is skew-adjoint — the physicist's self-adjoint \(H\) with \(A=-iH/\hbar\) — the propagator is unitary, so lengths, angles, and probabilities are conserved along the flow.

Units check. The argument \(tA\) must be dimensionless, so \([A]=\mathrm{time}^{-1}\): \(A\) is a rate. In the quantum form \(A=-iH/\hbar\), \([H/\hbar]=(\mathrm{J})/(\mathrm{J\,s})=\mathrm{s}^{-1}\), consistent. The eigenphases \(\lambda_{k}t/\hbar\) are (energy·time)/(action) \(=\) dimensionless radians, as a phase must be.

Limiting cases

\(A=0\): \(\exp(tA)=I\), the trivial flow — nothing moves; every point is a fixed point.
\(A\) diagonalizable, \(A=S\,\mathrm{diag}(\mu_{k})\,S^{-1}\): \(\exp(tA)=S\,\mathrm{diag}(e^{\mu_{k}t})\,S^{-1}\) — the flow decouples into scalar modes \(e^{\mu_{k}t}\).
\(A\) real skew-symmetric, \(A^{\mathsf T}=-A\): \(\exp(tA)\in SO(n)\) is a rotation; e.g. \(A=\omega\!\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\) gives rotation by \(\omega t\).
\(\mathrm{Re}\,\mu_{k}<0\) for all eigenvalues: \(\exp(tA)\to0\) as \(t\to\infty\) — asymptotically stable (dissipative) flow.
Small \(t\): \(\exp(tA)=I+tA+O(t^{2})\), recovering the infinitesimal generator \(A=\left.\frac{d}{dt}\right|_{0}U(t)\).

Breaks when

Time-dependent generator. For \(\dot x=A(t)x\), \(\exp\!\big(\int_{0}^{t}A\big)\) is not the propagator unless \([A(t_{1}),A(t_{2})]=0\) for all times; in general one needs the time-ordered (Dyson) exponential, and the group law \(U(s+t)=U(s)U(t)\) is replaced by a cocycle relation.
Non-commuting split generators. \(\exp\!\big(t(A+B)\big)\ne\exp(tA)\exp(tB)\) when \([A,B]\ne0\); the error is \(\tfrac{t^{2}}{2}[A,B]+\dots\) (Baker–Campbell–Hausdorff). Naïve operator splitting or Trotterization introduces \(O(t^{2})\) error per step.
Unbounded generator without a domain. On infinite-dimensional \(V\) with unbounded \(H\), the power series diverges; \(e^{-iHt/\hbar}\) exists only through the spectral calculus, and only a self-adjoint (not merely symmetric) \(H\) generates a unitary group.
Wrong metric / defective generator. If \(A\) is not skew-adjoint in the physical inner product (gain–loss systems, or a defective \(A\) with a Jordan block), the flow amplifies or decays; a Jordan block yields secular \(t\,e^{\mu t}\) growth rather than a bounded oscillation.

Failure modes

Exponentiating entrywise. Writing \((e^{tA})_{ij}=e^{tA_{ij}}\) instead of the operator series — wrong except for diagonal \(A\).
Splitting non-commuting exponents. Setting \(e^{A+B}=e^{A}e^{B}\) without checking \([A,B]=0\); this silently drops the BCH commutator terms.
Confusing symmetric with self-adjoint. For unbounded operators, symmetry \(\langle H\phi,\psi\rangle=\langle\phi,H\psi\rangle\) does not guarantee a unitary flow; self-adjointness (equal domains) is required.
Sign of \(i\). Using \(e^{+iHt/\hbar}\) for forward evolution — it evolves backward; the Schrödinger propagator is \(e^{-iHt/\hbar}\).
Assuming diagonalizability. Treating every \(A\) as \(S\Lambda S^{-1}\); defective matrices need the Jordan form and produce polynomial-times-exponential terms.
Forgetting \(\hbar\). Writing \(e^{-iHt}\) with \(H\) in joules gives a dimensionally broken phase; the argument must be \(Ht/\hbar\).

Discussion

The map \(t\mapsto\exp(tA)\) is a homomorphism from the additive group \((\mathbb{R},+)\) into the group of invertible operators: it is the prototypical one-parameter subgroup, and \(A\) is its Lie-algebra generator, recovered as the tangent at the identity, \(A=\dot U(0)\). This is the linear shadow of the exponential map of Lie theory, and it is why "symmetry generator" and "conserved observable" are the same object seen from two sides — the symmetry thread and the waves thread meet here.

When \(A=-iH/\hbar\) with \(H\) self-adjoint, the spectral theorem (step 8) turns evolution into a diagonal phase machine: each energy eigenstate merely rotates at angular frequency \(\lambda_{k}/\hbar\), and a general state is a superposition of these stationary rotations. Interference between the phases \(e^{-i\lambda_{k}t/\hbar}\) is the entire content of quantum dynamics — beats, Rabi flopping, and wave-packet spreading are all phase differences accumulating in time. Unitarity guarantees that no probability is created or destroyed while these phases wind.

Stone's theorem is the sharp converse of the derivation: every strongly continuous one-parameter unitary group \(U(t)\) on a Hilbert space is of the form \(e^{-iHt/\hbar}\) for a unique self-adjoint \(H=i\hbar\,\dot U(0)\), even when \(H\) is unbounded. Thus "self-adjoint generator" and "unitary flow" are not merely compatible but in bijection; the physical requirement of reversible, probability-conserving dynamics forces the Hamiltonian to be self-adjoint. Subtleties like the choice of self-adjoint extension (a particle in a box, the Aharonov–Bohm flux) are then physically real: distinct extensions are distinct boundary conditions, hence distinct physics, each generating its own unitary group.

Common misconceptions. The matrix exponential is not "\(A\) raised to a power" nor an entrywise operation; it is a convergent operator series whose defining property is \(\dot U=AU\). Unitarity is a statement about the generator and the metric, not about \(A\) being "nice"; and reversibility of the flow (\(U(-t)=U(t)^{-1}\)) holds for any constant generator, whereas norm-preservation is the special gift of skew-adjointness.

Worked examples

Planar rotational flow. \[ A=\begin{pmatrix}0&-\omega\\ \omega&0\end{pmatrix},\qquad \dot x=Ax,\quad x_{0}=\begin{pmatrix}1\\0\end{pmatrix} \]

Real skew-symmetric generator; expect an orthogonal (norm-preserving) flow. Symbolic first. A

\[ A^{2}=-\omega^{2}I\ \Rightarrow\ \exp(tA)=\sum_{n}\frac{(tA)^{n}}{n!}=\cos(\omega t)\,I+\frac{\sin(\omega t)}{\omega}\,A=\begin{pmatrix}\cos\omega t&-\sin\omega t\\ \sin\omega t&\cos\omega t\end{pmatrix} \]

Even/odd power split, exactly as \(e^{i\theta}=\cos\theta+i\sin\theta\); here \(A\) plays the role of \(i\). B

\[ \omega=3\ \mathrm{s^{-1}},\ t=0.5\ \mathrm{s}\Rightarrow\omega t=1.5\ \mathrm{rad};\quad x(0.5)=\begin{pmatrix}\cos1.5\\ \sin1.5\end{pmatrix}=\begin{pmatrix}0.0707\\ 0.997\end{pmatrix},\quad \lVert x\rVert=\sqrt{0.0707^{2}+0.997^{2}}=1.000 \]

Insert values only now; check the length is conserved. A

\[ x(0.5\,\mathrm{s})=(0.0707,\ 0.997),\qquad \lVert x\rVert=1 \]

Reading. The state has rotated by \(1.5\) rad in the plane; the orthogonal propagator preserved its unit length exactly, the real analogue of unitarity.

Units check. \(\omega t=(3\,\mathrm{s^{-1}})(0.5\,\mathrm{s})=1.5\) is dimensionless (radians); components of \(x\) inherit the dimensionless units of \(x_{0}\).

Qubit under a resonant drive (Rabi flop). \[ H=\frac{\hbar\Omega}{2}\,\sigma_{x},\qquad A=-\frac{i}{\hbar}H=-\frac{i\Omega}{2}\sigma_{x},\qquad U(t)=\exp\!\Big(-\tfrac{i\Omega t}{2}\sigma_{x}\Big) \]

Self-adjoint \(H\) (Pauli \(\sigma_{x}=\sigma_{x}^{\dagger}\)); the flow must be unitary. B

\[ \sigma_{x}^{2}=I\ \Rightarrow\ U(t)=\cos\!\Big(\tfrac{\Omega t}{2}\Big)I-i\sin\!\Big(\tfrac{\Omega t}{2}\Big)\sigma_{x},\qquad P_{0\to1}(t)=\big|\langle1|U|0\rangle\big|^{2}=\sin^{2}\!\Big(\tfrac{\Omega t}{2}\Big) \]

Same even/odd resummation; \(\langle1|\sigma_{x}|0\rangle=1\) picks out the sine term. B

\[ \Omega=2\pi\times5\ \mathrm{MHz},\ t=50\ \mathrm{ns}:\quad \tfrac{\Omega t}{2}=\pi\,(5\times10^{6}\,\mathrm{s^{-1}})(50\times10^{-9}\,\mathrm{s})=0.785\ \mathrm{rad},\quad P_{0\to1}=\sin^{2}(0.785)=0.500 \]

Insert values; \(P_{0}+P_{1}=\cos^{2}+\sin^{2}=1\) confirms unitarity. A

\[ P_{0\to1}(50\,\mathrm{ns})=\sin^{2}\!\Big(\tfrac{\Omega t}{2}\Big)=0.500\quad(\text{a }\pi/2\text{ pulse}) \]

Reading. A self-adjoint generator drives the qubit unitarily between \(|0\rangle\) and \(|1\rangle\); after \(50\ \mathrm{ns}\) the population is split \(50/50\) — an equal superposition, the standard \(\pi/2\) gate.

Units check. \(\Omega t/2\) is (rad/s)(s) \(=\) radians (dimensionless); \(P\) is a probability in \([0,1]\), and \(P_{0}+P_{1}=1\).

Problems

Show directly from the series that \(\det\exp(tA)=e^{t\,\mathrm{tr}\,A}\) (Liouville's formula). Hence state when the flow preserves phase-space volume.
Solution
Let \(\phi(t)=\det U(t)\). Differentiate using \(\dot U=AU\) and Jacobi's formula: \(\frac{d}{dt}\det U=\det U\,\mathrm{tr}(U^{-1}\dot U)=\det U\,\mathrm{tr}(U^{-1}AU)=\det U\,\mathrm{tr}\,A\). So \(\dot\phi=(\mathrm{tr}\,A)\phi\), \(\phi(0)=1\), giving \(\phi(t)=e^{t\,\mathrm{tr}\,A}\). Volume is preserved (\(\det=1\)) iff \(\mathrm{tr}\,A=0\); a skew-adjoint generator (which has purely imaginary trace, and traceless if real) qualifies — Liouville's theorem for unitary/Hamiltonian flows.
For \(A=\left(\begin{smallmatrix}2&1\\0&2\end{smallmatrix}\right)\) (a Jordan block, \(\mu=2\)) compute \(\exp(tA)\) and describe the long-time behaviour.
Solution
Split \(A=2I+N\) with \(N=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\), \(N^{2}=0\); \(2I\) and \(N\) commute so \(\exp(tA)=e^{2t}\exp(tN)=e^{2t}(I+tN)=e^{2t}\left(\begin{smallmatrix}1&t\\0&1\end{smallmatrix}\right)\). The off-diagonal entry \(t\,e^{2t}\) shows secular polynomial-times-exponential growth: defective (non-diagonalizable) \(A\) does not give pure exponentials, and here the flow is unbounded and unstable.
A damped oscillator has \(\ddot q+2\gamma\dot q+\omega_{0}^{2}q=0\). Write it as \(\dot x=Ax\) with \(x=(q,\dot q)^{\mathsf T}\), find the eigenvalues of \(A\), and give the condition for \(\exp(tA)\to0\).
Solution
\(A=\left(\begin{smallmatrix}0&1\\-\omega_{0}^{2}&-2\gamma\end{smallmatrix}\right)\). Characteristic equation \(\mu^{2}+2\gamma\mu+\omega_{0}^{2}=0\Rightarrow\mu_{\pm}=-\gamma\pm\sqrt{\gamma^{2}-\omega_{0}^{2}}\). Since \(\mathrm{tr}\,A=-2\gamma\) and \(\det A=\omega_{0}^{2}\), both eigenvalues have negative real part iff \(\gamma>0\) and \(\omega_{0}^{2}>0\). Then \(\mathrm{Re}\,\mu_{\pm}<0\) and \(\exp(tA)\to0\): asymptotic stability. (Underdamped for \(\gamma<\omega_{0}\), overdamped for \(\gamma>\omega_{0}\).)
Let \(H=\frac{\hbar\Omega}{2}\sigma_{z}\). Compute \(U(t)=e^{-iHt/\hbar}\) and evolve \(|\psi(0)\rangle=\frac{1}{\sqrt2}(|0\rangle+|1\rangle)\). What is \(\langle\sigma_{x}\rangle(t)\)?
Solution
\(\sigma_{z}\) is diagonal so \(U(t)=\mathrm{diag}(e^{-i\Omega t/2},e^{+i\Omega t/2})\). Then \(|\psi(t)\rangle=\frac{1}{\sqrt2}(e^{-i\Omega t/2}|0\rangle+e^{+i\Omega t/2}|1\rangle)\). With \(\langle\sigma_{x}\rangle=2\,\mathrm{Re}(c_{0}^{*}c_{1})\) and \(c_{0}^{*}c_{1}=\tfrac12 e^{+i\Omega t}\), we get \(\langle\sigma_{x}\rangle(t)=\cos(\Omega t)\). The Bloch vector precesses about \(z\) at angular frequency \(\Omega\); the norm \(\langle\psi|\psi\rangle=1\) at all times (unitary).
Prove the group law fails for non-commuting generators to leading order: show \(e^{tA}e^{tB}=e^{t(A+B)+\frac{t^{2}}{2}[A,B]+O(t^{3})}\), and evaluate the discrepancy for \(A=\sigma_{x},\,B=\sigma_{y}\) at small \(t\).
Solution
Expand each factor: \(e^{tA}e^{tB}=(I+tA+\tfrac{t^2}{2}A^2)(I+tB+\tfrac{t^2}{2}B^2)+O(t^3)=I+t(A+B)+\tfrac{t^2}{2}(A^2+2AB+B^2)+O(t^3)\). Meanwhile \(e^{t(A+B)}=I+t(A+B)+\tfrac{t^2}{2}(A+B)^2+O(t^3)\) with \((A+B)^2=A^2+AB+BA+B^2\). Subtracting, the \(t^2\) difference is \(\tfrac{t^2}{2}(AB-BA)=\tfrac{t^2}{2}[A,B]\), matching BCH. For \(A=\sigma_x,B=\sigma_y\), \([\sigma_x,\sigma_y]=2i\sigma_z\), so the leading discrepancy is \(t^2\,i\sigma_z\) — a rotation about \(z\) that the commuting formula misses.