Proof of Basic Properties of the State-Transition Matrix

Computations with real numbers are those of The Real Numbers as a Type. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; unless stated otherwise it is the interval $I$ of Rules for Derivatives of Vector- and Matrix-Valued Maps. By The State-Transition Matrix of a Linear System (ii) and The State-Transition Matrix of a Linear System (iii), for every $s\in[a,b]$ the map $\Phi(\cdot,s)$ is differentiable with derivative $A(t)\Phi(t,s)$ at every $t\in[a,b]$, and $\Phi(s,s)=I_{n}$.

Proof of (i). Let $s\in[a,b]$ and $\xi\in\mathbb{R}^{n}$, and let $x_{0}:[a,b]\to\mathbb{R}^{n}$, $t\mapsto\Phi(t,s)\xi$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (x), with $\Phi(\cdot,s)$ for $A$ and $\xi$ for $z$, $x_{0}$ is differentiable at every $t\in[a,b]$, hence differentiable, with derivative $\bigl(A(t)\Phi(t,s)\bigr)\xi=A(t)\bigl(\Phi(t,s)\xi\bigr)$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v); and $x_{0}(s)=I_{n}\xi=\xi$. Let $x:[a,b]\to\mathbb{R}^{n}$ be differentiable with $x'(t)=A(t)x(t)$ for every $t\in[a,b]$ and $x(s)=\xi$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (iv) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v), $y=x-x_{0}$ is differentiable with $y'(t)=A(t)x(t)-A(t)x_{0}(t)=A(t)y(t)$, and $y(s)=\xi-\xi=0$ (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii)). By Existence and Uniqueness of the State-Transition Matrix of a Linear System (i), $y(t)=0$ for every $t$, so $x=x_{0}$. For $q\in\mathbb{N}$ and $\Xi\in\mathcal{M}_{n\times q}(\mathbb{R})$, the map $X_{0}:t\mapsto\Phi(t,s)\Xi=\bigl(I_{n}\Phi(t,s)\bigr)\Xi$ is differentiable by Rules for Derivatives of Vector- and Matrix-Valued Maps (x), with $I_{n}$ for $M$ and $\Xi$ for $N$, with derivative $\bigl(I_{n}A(t)\Phi(t,s)\bigr)\Xi=A(t)\bigl(\Phi(t,s)\Xi\bigr)$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i), and $X_{0}(s)=I_{n}\Xi=\Xi$; if $X$ is as in (i), then $X-X_{0}$ is differentiable by Rules for Derivatives of Vector- and Matrix-Valued Maps (iv), with derivative $A(t)\bigl(X(t)-X_{0}(t)\bigr)$ by Products of Real Matrices with Zero Matrices, Negatives and Differences (iii), and vanishes at $s$, so $X=X_{0}$ by the matrix half of Existence and Uniqueness of the State-Transition Matrix of a Linear System (i).

Proof of (ii). Let $r,s\in[a,b]$. By the matrix half of (i), with $n$ for $q$, $r$ for $s$ and $\Phi(r,s)$ for $\Xi$, the map $t\mapsto\Phi(t,r)\Phi(r,s)$ is the only differentiable map $X:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ with $X'(t)=A(t)X(t)$ for every $t$ and $X(r)=\Phi(r,s)$. The map $\Phi(\cdot,s)$ is such a map. Hence $\Phi(t,r)\Phi(r,s)=\Phi(t,s)$ for every $t\in[a,b]$.

Proof of (iii). Let $s,t\in[a,b]$. By (ii) and The State-Transition Matrix of a Linear System (iii), $\Phi(t,s)\Phi(s,t)=\Phi(t,t)=I_{n}$ and $\Phi(s,t)\Phi(t,s)=\Phi(s,s)=I_{n}$. So $\Phi(s,t)$ is an inverse of $\Phi(t,s)$ by Inverse Matrix and Invertible Real Square Matrix (i); $\Phi(t,s)$ is invertible, and $\Phi(t,s)^{-1}=\Phi(s,t)$ by Inverse Matrix and Invertible Real Square Matrix (ii).

Proof of (v). Let $W=\Phi(\cdot,a):[a,b]\to\mathcal{M}_{n}(\mathbb{R})$; it is differentiable with $W'(s)=A(s)\Phi(s,a)$, and by (iii) $W(s)$ is invertible with $W(s)^{-1}=\Phi(a,s)$, for every $s\in[a,b]$. By Continuity and Derivative of the Inverse of a Matrix-Valued Map (iii), with $n$ for $p$ and $W$ for $A$, the map $W^{-1}:s\mapsto\Phi(a,s)$ is differentiable at every $s\in[a,b]$, with derivative, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i), Products of Real Matrices with Zero Matrices, Negatives and Differences (ii), (ii), The State-Transition Matrix of a Linear System (iii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv),

$$-\Phi(a,s)\bigl(A(s)\Phi(s,a)\bigr)\Phi(a,s)=-\Phi(a,s)A(s)\bigl(\Phi(s,a)\Phi(a,s)\bigr)=-\Phi(a,s)A(s).$$

Let $t\in[a,b]$. By (ii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv), the map $s\mapsto\Phi(t,s)$ is $s\mapsto\bigl(\Phi(t,a)W^{-1}(s)\bigr)I_{n}$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (x), with $\Phi(t,a)$ for $M$ and $I_{n}$ for $N$, it is differentiable at every $s\in[a,b]$, hence differentiable, with derivative

$$\Phi(t,a)\bigl(-\Phi(a,s)A(s)\bigr)=-\bigl(\bigl(\Phi(t,a)\Phi(a,s)\bigr)A(s)\bigr)=-\bigl(\Phi(t,s)A(s)\bigr),$$

by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i), Products of Real Matrices with Zero Matrices, Negatives and Differences (ii) and (ii).

Proof of (iv). Let $U=\Phi(\cdot,a)$ and $V:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $s\mapsto\Phi(a,s)$. Both are differentiable, by The State-Transition Matrix of a Linear System (ii) and (v), hence continuous at every point of $[a,b]$ relative to $[a,b]$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (i), and continuous on $[a,b]$ by Continuous Map Between Metric Spaces (ii). Let $d$ be the product metric on $\mathbb{R}\times\mathbb{R}$, so that $(\mathbb{R}\times\mathbb{R},d)$ is a metric space by The Product Metric is a Metric (i), and let $S=[a,b]\times[a,b]$. The map $\tilde{U}:S\to\mathcal{M}_{n}(\mathbb{R})$, $(t,s)\mapsto U(t)$, is continuous on $S$: for $(t_{0},s_{0})\in S$ and a real $\varepsilon>0$, let $\delta>0$ be given by Continuous Map Between Metric Spaces (i) for $U$ at $t_{0}$ and $\varepsilon$; if $(t,s)\in S$ and $d\bigl((t_{0},s_{0}),(t,s)\bigr)<\delta$, then $d_{\mathbb{R}}(t_{0},t)<\delta$ by The Product Metric is a Metric (ii), so $d_{n}\bigl(\tilde{U}(t,s),\tilde{U}(t_{0},s_{0})\bigr)=d_{n}\bigl(U(t),U(t_{0})\bigr)<\varepsilon$. In the same way $\tilde{V}:S\to\mathcal{M}_{n}(\mathbb{R})$, $(t,s)\mapsto V(s)$, is continuous on $S$. By (ii), $\Phi(t,s)=\Phi(t,a)\Phi(a,s)=\tilde{U}(t,s)\tilde{V}(t,s)$ for $(t,s)\in S$, so $\Phi$ is continuous on $S$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), with $(\mathbb{R}\times\mathbb{R},d)$ for its $(X,d)$.

Proof of (vii). The map $t\mapsto A(t)^{\top}$, the constant map with value $-I_{n}$ and their product $t\mapsto(-I_{n})A(t)^{\top}=-A(t)^{\top}$ (Products of Real Matrices with Zero Matrices, Negatives and Differences (ii), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv)) are continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v); so $-A^{\top}$ is continuous on $[a,b]$ and has a state-transition matrix on $[a,b]$ by The State-Transition Matrix of a Linear System (i). Let $\Theta:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be the map $(t,s)\mapsto\Phi(s,t)^{\top}$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)), and let $s\in[a,b]$. By (v), with $s$ for $t$, the map $t\mapsto\Phi(s,t)$ is differentiable with derivative $-\bigl(\Phi(s,t)A(t)\bigr)$ at every $t\in[a,b]$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (viii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vii), $\Theta(\cdot,s)$, $t\mapsto\Phi(s,t)^{\top}$, is differentiable with derivative

$$\Bigl(-\bigl(\Phi(s,t)A(t)\bigr)\Bigr)^{\top}=-A(t)^{\top}\Phi(s,t)^{\top}=\bigl(-A(t)^{\top}\bigr)\Theta(t,s)$$

at every $t\in[a,b]$, using Products of Real Matrices with Zero Matrices, Negatives and Differences (ii); and $\Theta(s,s)=I_{n}^{\top}=I_{n}$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vii). By the uniqueness in Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii), with $-A^{\top}$ for $A$, and by The State-Transition Matrix of a Linear System (i), $\Theta=\Phi_{-A^{\top}}$.

Proof of (viii). Let $c,d\in[a,b]$ with $c<d$. By Basic Facts about Intervals of the Real Line (ii) and Basic Facts about Intervals of the Real Line (v), $[c,d]\subseteq[a,b]$ and $[c,d]$ is an interval containing the two distinct points $c$ and $d$. For $t\in[c,d]$ and a real $\varepsilon>0$, the $\delta$ that Continuous Map Between Metric Spaces (i) gives for $A$ at $t$ relative to $[a,b]$ also serves for $A|_{[c,d]}$ at $t$ relative to $[c,d]$, as $A|_{[c,d]}(u)=A(u)$ for $u\in[c,d]$; so $A|_{[c,d]}$ is continuous on $[c,d]$. Let $\Gamma:[c,d]\times[c,d]\to\mathcal{M}_{n}(\mathbb{R})$ be the restriction of $\Phi$, and let $s\in[c,d]$. For $i,j\in[n]$, the entry map of $\Gamma(\cdot,s)$ in row $i$ and column $j$ is the restriction to $[c,d]$ of that of $\Phi(\cdot,s)$, which is differentiable at every $t\in[c,d]$ relative to $[a,b]$ by Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii); by Restriction Stability of Continuity and of the Derivative (ii), with $[a,b]$ for $I$ and $[c,d]$ for $J$, it is differentiable at $t$ relative to $[c,d]$ with the same derivative. Hence, by Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), $\Gamma(\cdot,s)$ is differentiable relative to $[c,d]$, with derivative $A(t)\Phi(t,s)=A|_{[c,d]}(t)\,\Gamma(t,s)$ at every $t\in[c,d]$; and $\Gamma(s,s)=I_{n}$. By the uniqueness in Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii), with $c$, $d$ and $A|_{[c,d]}$ for $a$, $b$ and $A$, and by The State-Transition Matrix of a Linear System (i), $\Gamma=\Phi_{A|_{[c,d]}}$.

Proof of (vi). Case $s\le t$. We show this case for every map $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ continuous on $[a,b]$ with $\lVert A(r)\rVert\le K$ for every $r\in[a,b]$. If $s=t$, then $\lVert\Phi(s,s)\rVert=\lVert I_{n}\rVert=1=\exp(0)$ by The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (vi). Let $s<t$; then $s<b$. By (viii), with $s$ and $b$ for $c$ and $d$, $A|_{[s,b]}$ is continuous on $[s,b]$ and $\Gamma=\Phi_{A|_{[s,b]}}$ is the restriction of $\Phi$ to $[s,b]\times[s,b]$. Let $O:[s,b]\to\mathcal{M}_{n}(\mathbb{R})$ be the constant map with value $0_{n}$, continuous on $[s,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), and let $F:[s,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the map of Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (i), with $s$ for $a$, $n$ for $q$, and $A|_{[s,b]}$, $O$ and $O$ for $A$, $B$ and $G$: $F(r,Z)=A(r)Z+Z0_{n}+0_{n}=A(r)Z$ (Products of Real Matrices with Zero Matrices, Negatives and Differences (i) and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)). By Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (iv), with $s$ for $a$, $n$ for $q$, and $A|_{[s,b]}$, $O$ and $O$ for $A$, $B$ and $G$, and with $K$ and $0$ for $K_{A}$ and $K_{B}$ (as $\lVert0_{n}\rVert=0$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iv)), $F$ is continuous on $[s,b]\times\mathcal{M}_{n}(\mathbb{R})$ and $\lVert F(r,Z)-F(r,Z')\rVert\le K\lVert Z-Z'\rVert$ for $r\in[s,b]$ and $Z,Z'\in\mathcal{M}_{n}(\mathbb{R})$. By The State-Transition Matrix of a Linear System (ii) and The State-Transition Matrix of a Linear System (iii), applied on $[s,b]$ to $A|_{[s,b]}$, the map $\Gamma(\cdot,s)$ is differentiable relative to $[s,b]$ with derivative $A(r)\Gamma(r,s)=F\bigl(r,\Gamma(r,s)\bigr)$ at every $r\in[s,b]$, and $\Gamma(s,s)=I_{n}$; by Rules for Derivatives of Vector- and Matrix-Valued Maps (ix), with $[s,b]$ for $I$, the constant map $r\mapsto0_{n}$ on $[s,b]$ is differentiable with derivative $0_{n}=F(r,0_{n})$. By Solution of an Initial Value Problem on $[a,c]$ (ii) and Solution of an Initial Value Problem on $[a,c]$ (iii), with $s$ for $a$ and $b$ for $c$, these are the solutions of the initial value problems of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) on $[s,b]$ with right-hand side $F$ and initial values $I_{n}$ and $0_{n}$. By The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (iv), with $s$ for $a$, $n$ for $p$ and $q$, and $K$ for $L$,

$$\lVert\Phi(t,s)\rVert=\lVert\Gamma(t,s)-0_{n}\rVert\le\lVert I_{n}-0_{n}\rVert\exp\bigl(K(t-s)\bigr)=\exp\bigl(K|t-s|\bigr),$$

using Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii) (for $X-0_{n}=X$), The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (vi) and $t-s=|t-s|$.

Case $t<s$. By (vii), $-A^{\top}$ is continuous on $[a,b]$, and for $r\in[a,b]$, $\lVert-A(r)^{\top}\rVert=\lVert A(r)^{\top}\rVert=\lVert A(r)\rVert\le K$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (v) and The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (v). The first case, applied to $-A^{\top}$ with $t$ and $s$ interchanged, and (vii) give

$$\lVert\Phi(t,s)\rVert=\lVert\Phi(t,s)^{\top}\rVert=\lVert\Phi_{-A^{\top}}(s,t)\rVert\le\exp\bigl(K|s-t|\bigr)=\exp\bigl(K|t-s|\bigr),$$

the first equality by The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (v). This proves (vi).