Basic Properties of the State-Transition Matrix

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b\in\mathbb{R}$ with $a<b$, let $n\in\mathbb{N}$, let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be continuous on $[a,b]$, for $d_{\mathbb{R}}$ on $[a,b]$ and $d_{n}$ on $\mathcal{M}_{n}(\mathbb{R})$, and let $\Phi=\Phi_{A}$ be the state-transition matrix of $A$ on $[a,b]$. Differentiability and derivatives of maps from $[a,b]$ into $\mathbb{R}^{n}$ or into $\mathcal{M}_{n\times q}(\mathbb{R})$ ($q\in\mathbb{N}$) are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii), taken relative to $[a,b]$.

(i) Let $s\in[a,b]$ and $\xi\in\mathbb{R}^{n}$. The map $[a,b]\to\mathbb{R}^{n}$, $t\mapsto\Phi(t,s)\xi$, is differentiable, with derivative $A(t)\bigl(\Phi(t,s)\xi\bigr)$ at every $t\in[a,b]$, and it is the only differentiable map $x:[a,b]\to\mathbb{R}^{n}$ with $x'(t)=A(t)x(t)$ for every $t\in[a,b]$ and $x(s)=\xi$. Likewise, for $q\in\mathbb{N}$ and $\Xi\in\mathcal{M}_{n\times q}(\mathbb{R})$, the map $t\mapsto\Phi(t,s)\Xi$ is differentiable, with derivative $A(t)\bigl(\Phi(t,s)\Xi\bigr)$ at every $t\in[a,b]$, and it is the only differentiable map $X:[a,b]\to\mathcal{M}_{n\times q}(\mathbb{R})$ with $X'(t)=A(t)X(t)$ for every $t\in[a,b]$ and $X(s)=\Xi$.

(ii) $\Phi(t,r)\Phi(r,s)=\Phi(t,s)$ for all $r,s,t\in[a,b]$.

(iii) For all $s,t\in[a,b]$, $\Phi(t,s)$ is invertible, and $\Phi(t,s)^{-1}=\Phi(s,t)$.

(iv) $\Phi$ is continuous on $[a,b]\times[a,b]$, for the product metric of $d_{\mathbb{R}}$ and $d_{\mathbb{R}}$ on $\mathbb{R}\times\mathbb{R}$, a metric by The Product Metric is a Metric (i), and for $d_{n}$ on $\mathcal{M}_{n}(\mathbb{R})$.

(v) For every $t\in[a,b]$, the map $[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $s\mapsto\Phi(t,s)$, is differentiable, with derivative $-\bigl(\Phi(t,s)A(s)\bigr)$ at every $s\in[a,b]$.

(vi) Here $\lVert\cdot\rVert$ is the operator norm, as in Euclidean Space and Real Matrices: Standing Notation (vi), and $\exp$ is the exponential function. Let $K\in\mathbb{R}$ with $K\ge0$ and $\lVert A(r)\rVert\le K$ for every $r\in[a,b]$. Then $\lVert\Phi(t,s)\rVert\le\exp\bigl(K|t-s|\bigr)$ for all $s,t\in[a,b]$.

(vii) The map $-A^{\top}:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto-A(t)^{\top}$, is continuous on $[a,b]$, and its state-transition matrix $\Phi_{-A^{\top}}$ on $[a,b]$ satisfies $\Phi_{-A^{\top}}(t,s)=\Phi(s,t)^{\top}$ for all $s,t\in[a,b]$.

(viii) Let $c,d\in[a,b]$ with $c<d$. The restriction $A|_{[c,d]}:[c,d]\to\mathcal{M}_{n}(\mathbb{R})$ is continuous on $[c,d]$, and its state-transition matrix $\Phi_{A|_{[c,d]}}$ on $[c,d]$ satisfies $\Phi_{A|_{[c,d]}}(t,s)=\Phi(t,s)$ for all $s,t\in[c,d]$.