The Matrix Exponential

Matrix Exponential

The most studied matrix function, used in solving differential equations.

eA

The matrix exponential of matrix A.

Naive Approach

Applying the exponential function to each entry of A.

Matrix Identity

(eA)2 = e2A does not hold.

Power Series Representation

If A is 1x1, (At)k eAt = k! / ∞ ∑ k=0 (At)k / k!.

Radius of Convergence

The series (1) converges for any A and any t (i.e., has radius of convergence equal to +∞).

Useful Properties of eA

d/dt eAt = AeAt = eAtA.

Commuting Matrices

eA+B = eAeB if and only if AB = BA.

Jordan Form Representation

eAt = XeJtX−1 = X diag(eJ1t, eJ2t, ..., eJpt)X−1.

Jordan Block

A square matrix with eigenvalue λ and index of nilpotency mi.

Differential Equations

Linear homogeneous initial value problem: ˙y(t) = Ay(t), y(0) = y0.

Solution to Differential Equations

y(t) = eAty0.

Inhomogeneous System

˙y(t) = Ay(t) + f(t, y), y(0) = y0, y ∈ Cn, A ∈ Cn×n.

Explicit Formula

y(t) = eAty0 + ∫[0,t] eA(t-s)f(s, y) ds.

Higher Order Equations

nth order homogeneous differential equation with constant coefficients.

Companion Form

Matrix A is the companion form of the scalar polynomial p(λ).

Distinct Eigenvalues

Jordan canonical form of A is J = diag(J1(λ1), ..., Jp(λp)).

Solution to nth Order Equation

x(t) = eT1X diag(eJ1t, ..., eJpt)X−1b.

Sylvester's Differential Equation

˙Y (t) = AY (t) + Y (t)B, Y (0) = Y0, has solution Y (t) = eAtY0eBt.

eA Properties

(i) e0 = I, (ii) (eA)∗ = eA∗, (iii) (eAt)−1 = e−At.

Unitary Matrix

A ∈ Cn×n is unitary if and only if A = eiH for some Hermitian H.

Eigenvalue and Eigenvector

If A has eigenvalue λ and eigenvector x, then (eλ, x) is an eigenpair for eA.

Determinant of eA

det(eA) = exp(trace(A)).

Constant Row Sums

If Au = αu, then eA has constant row sums eα.

Idempotent Matrix

(i) Minimal polynomial of P is P(x) = x(x-1), (ii) eP = I + (e - 1)P.