First-Order Systems and the Matrix Exponential

1Introduction

For a constant matrix $$A\in K^{n\times n}$$A\in\mathbb K^{n\times n}, where $$\mathbb{K}=\mathbb{R}$$\mathbb K=\mathbb R or $$\mathbb{C}$$\mathbb C, this lecture constructs the unique solution of

as $$\mathit{x}\left(t\right)=e^{A(t-t_0)}{x}_0$$\boldsymbol{x}(t)=e^{A(t-t_0)}\boldsymbol{x}_0.

2Convergence and Termwise Differentiation

Define

For any submultiplicative matrix norm,

The scalar majorant converges and is uniform on bounded time intervals. The differentiated series has the same type of majorant; termwise differentiation is therefore valid and gives

For the maximum row-sum norm,

we have

Thus this norm is submultiplicative and supplies the preceding majorant.

3Invertibility, Uniqueness, and the Composition (Group) Law

Since $$e^{A0}=I$$e^{A0}=I and

the product equals $$I$$I for every $$t$$t. Reversing the factors gives the other inverse identity, so

The proposed solution satisfies the equation and initial condition. If $$\mathit{y}$$\boldsymbol y is any other solution, then

which proves uniqueness. Applying uniqueness to two matrix solutions yields

4Computation and Real Matrices

If $$A=PD{P}^{-1}$$A=PDP^{-1}, then

For a real matrix diagonalizable only over $$\mathbb{C}$$\mathbb C, conjugate complex solutions combine through their real and imaginary parts to form real solutions. A nonorthogonal eigenbasis still gives the exact evolution, although eigenvalues alone need not describe transient Euclidean norm growth.

For

the identity $$A^2=-{\omega }^{2}I$$A^2=-\omega^2I gives

Because $$A^{\mathsf{T}}=-A$$A^{\mathsf T}=-A, this evolution preserves the Euclidean norm.
It rotates counterclockwise for $$\omega >0$$\omega>0 and clockwise for $$\omega <0$$\omega<0, with angular speed $$\left|\omega \right|$$|\omega|. This norm preservation follows from skew-symmetry; purely imaginary eigenvalues alone do not imply norm preservation for an arbitrary matrix.

5Reduction of Higher-Order Equations

Introducing $$\mathit{x}=(y,y^{\prime },\dots ,y^{(m-1)}{)}^{\mathsf{T}}$$\boldsymbol{x}=(y,y',\ldots,y^{(m-1)})^{\mathsf T} converts

into a first-order companion system. Its initial vector contains the values of $$y$$y and its first $$m-1$$m-1 derivatives. Constant coefficients produce a constant matrix; variable coefficients produce a time-dependent matrix.

Explicitly,

6Variation of Constants

For continuous $$\mathit{b}$$\boldsymbol b,

has the solution

Differentiation verifies both the equation and initial condition.
Indeed, writing $$\mathit{x}=e^{A(t-t_0)}\mathit{u}\left(t\right)$$\boldsymbol{x}=e^{A(t-t_0)}\boldsymbol u(t) gives

and integration from $$t_0$$t_0 to $$t$$t yields the formula. The difference of two solutions satisfies the homogeneous system, so uniqueness follows from the homogeneous result.

7Boundary for Time-Dependent Matrices

For continuous $$A\left(t\right)$$A(t), the corresponding initial-value problem has a unique solution. However, the expression

is valid as the evolution operator when $$A\left(t\right)A\left(s\right)=A\left(s\right)A\left(t\right)$$A(t)A(s)=A(s)A(t) for every pair of times. Without this commutativity, chronological order matters and a time-ordered construction is required. The formula $$e^{A(t-t_0)}$$e^{A(t-t_0)} in this lecture is restricted to constant $$A$$A.

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