4.1 The space of solutions
Theorem 4.1.1. The solutions of the homogeneous system \(x' = A(t)x\) on \(I\) form a vector space of dimension \(n\). For any \(t_0 \in I\), the map \(x \mapsto x(t_0)\) is an isomorphism from this space onto \(\mathbb R^n\).
Proof. A linear combination of solutions is a solution, since the equation is linear, so the solutions form a vector space \(S\). The evaluation map \(E\colon x \mapsto x(t_0)\) is linear. It is onto, because for every \(x_0\) the initial value problem has a solution on \(I\), and one-to-one, because a solution with \(x(t_0) = 0\) is identically zero by uniqueness (the zero function is a solution). So \(E\) is an isomorphism and \(\dim S = n\). □
A consequence that is easy to overlook: solutions \(x_1, \dots , x_k\) are linearly independent as functions if and only if their values \(x_1(t), \dots , x_k(t)\) are linearly independent vectors at any one time \(t\), and then at every time. For solutions, independence at one instant is independence for ever.
Definition 4.1.2. A fundamental matrix is an \(n \times n\) matrix \(\Phi (t)\) whose columns are \(n\) linearly independent solutions, so that \(\Phi ' = A(t)\Phi \) and \(\det \Phi (t) \neq 0\). The transition matrix (or principal fundamental matrix at \(s\)) is \(\Phi (t, s) = \Phi (t)\Phi (s)^{-1}\).
Every solution is \(x(t) = \Phi (t)c\) for a constant vector \(c\), and the solution with \(x(s) = x_s\) is \(x(t) = \Phi (t, s)x_s\). The transition matrix does not depend on which fundamental matrix is used, since any two differ by a constant invertible factor, \(\Psi (t) = \Phi (t)C\), which cancels in \(\Psi (t)\Psi (s)^{-1}\).
Proposition 4.1.3. The transition matrix satisfies \(\Phi (t, t) = I\), \(\Phi (t, s)\Phi (s, r) = \Phi (t, r)\), \(\Phi (t, s)^{-1} = \Phi (s, t)\), and \[ \frac {\partial }{\partial t}\Phi (t, s) = A(t)\Phi (t, s), \qquad \frac {\partial }{\partial s}\Phi (t, s) = -\Phi (t, s)A(s) . \]
Proof. The first three follow from \(\Phi (t, s) = \Phi (t)\Phi (s)^{-1}\). The first derivative is \(\Phi '(t)\Phi (s)^{-1} = A(t)\Phi (t)\Phi (s)^{-1}\). For the second, differentiating \(\Phi (s)\Phi (s)^{-1} = I\) gives \[ \frac {d}{ds}\Phi (s)^{-1} = -\Phi (s)^{-1}\Phi '(s)\Phi (s)^{-1} = -\Phi (s)^{-1}A(s) , \] and multiplying on the left by \(\Phi (t)\) gives the result. □
The composition law is the flow property of Theorem 3.5.1 for linear systems: the state at time \(r\) is carried to time \(s\) and then to time \(t\).
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