Variation of constants for a finite-matrix semilinear system

Given: The finite matrix, interval, curve, and semilinear differential equation in the Statement.

First-order systems, initial value problems, and solutions on intervals: a solution is differentiable and satisfies its differential equation at every time in its interval.

A bounded linear operator between normed spaces, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, and Every finite-dimensional normed space is Banach: a finite matrix is a bounded operator on the complete finite-dimensional space ${\mathbb{C}}^d$, so The exponential series of a bounded operator applies and gives $(e^{rA}{)}^{\mathrm{\prime }}=A{e}^{rA}=e^{rA}A$ and $e^{rA}{e}^{qA}=e^{(r+q)A}$.

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: a curve in ${\mathbb{C}}^d$ and its integral are interpreted coordinatewise in the real vector space ${\mathbb{R}}^{2d}$; an integral with equal limits is zero.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: every continuous real coordinate of a finite-dimensional vector curve on a closed bounded interval is integrable.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: for $s<t$ and a differentiable finite-dimensional real vector curve with integrable derivative, $z(t)-z(s)={\int }_s^t{z}^{\mathrm{\prime }}(r)dr$.

Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes and Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: matrix multiplication is a finite sum of scalar products, so its derivative obeys the product rule.

Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$: a fixed finite-dimensional linear map commutes with componentwise vector integration.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $y=(y_0,\dots ,y_{d-1})$, set ${\mathrm{\Phi }}_d(y):=(\mathrm{Re}{y}_0,\mathrm{Im}{y}_0,\dots ,\mathrm{Re}{y}_{d-1},\mathrm{Im}{y}_{d-1})$. Applying the scalar coordinate map componentwise identifies ${\mathbb{C}}^d$ real-linearly with ${\mathbb{R}}^{2d}$, and $\mathrm{\parallel }{\mathrm{\Phi }}_d(y){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}((\mathrm{Re}{y}_j{)}^2+(\mathrm{Im}{y}_j{)}^2)={\sum }_{j<d}\mathrm{\mid }{y}_j{\mathrm{\mid }}^2=\mathrm{\parallel }y{\mathrm{\parallel }}_2^2$. It carries each real-time difference quotient to the corresponding realified quotient and preserves its limit, while matrix-vector multiplication becomes the corresponding real coordinate arithmetic. Thus the real differentiation and integration suppliers apply without changing the norm.