The flow and variational equation of a linear system: Phi(t,a)=e^{tA}a

Given: $n\ge 1$, a matrix $A\in M_n(\mathbb{R})$ and the linear field $F(x)=Ax$. Coordinates are zero-based; a matrix acts on a column vector by $(Ax{)}_i={\sum }_{j<n}{A}_{ij}{x}_j$. Vector norms are Euclidean (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), and $\mathrm{\parallel }A\mathrm{\parallel }$ is the subordinate Euclidean operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Euclidean ${\mathbb{R}}^n$ is complete ($\mathbb{R}$ and ${\mathbb{R}}^n$ for $n\ge 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $\mathbb{R}$). Rowwise Cauchy-Schwarz gives $\mathrm{\parallel }Ax{\mathrm{\parallel }}_2^2\le ({\sum }_{i,j<n}{A}_{ij}^2)\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$, so $A$ is bounded and $F$ is globally Lipschitz (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation). The exponential series defines a $C^{\mathrm{\infty }}$ group $t\mapsto e^{tA}$ with $\frac{d}{dt}{e}^{tA}=A{e}^{tA}=e^{tA}A$, $e^0=I_n$ and $e^{(s+t)A}=e^{sA}{e}^{tA}$ (The exponential series of a bounded operator, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

Every initial value problem of $x^{\mathrm{\prime }}=Ax$ has a unique maximal solution, and every solution through the same initial data is a restriction of it (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, Extensions, maximal solutions, maximal intervals, and global solutions, First-order systems, initial value problems, and solutions on intervals, The local flow of an autonomous first-order system).

The variational equation along a solution $x$ of an autonomous $C^1$ field is the linear matrix ODE $Y^{\mathrm{\prime }}=DF(x(t))Y$; for the linear field it is $Y^{\mathrm{\prime }}=AY$, and it has a unique solution on every compact interval for every prescribed initial matrix at a time in that interval (The variational equation along an ODE solution, Linear matrix ODEs have unique global solutions on a fixed interval). For $a_0\in {\mathbb{R}}^n$, $R,T>0$, $M:={\sup }_{\bar{B}(a_0,R)}\mathrm{\parallel }Ax{\mathrm{\parallel }}_2$ and $0<\rho <R-MT$, the smooth-dependence theorem identifies $D_a\mathrm{\Phi }(t,a)h$ with $Y_a(t)h$ for $a\in B(a_0,\rho )$ and $t\in [0,T]$, with the derivative bound $L=\mathrm{\parallel }A\mathrm{\parallel }$ and the trapping bound $\mathrm{\parallel }\mathrm{\Phi }(t,a)-a_0{\mathrm{\parallel }}_2\le \rho +Mt<R$ (C¹ dependence of an autonomous flow on the initial condition, and the variational equation).