3. Appendix to §2

Here we show how local solutions to the system dx/dt = A(t) x on two overlapping intervals can be combined to give a solution on their union, thus filling in the gap in the proof of the Proposition in §2. We in fact get a slightly stronger result, based on the following

Lemma. for any s₀ in the domain of A(t), there exists a neighbourhood N of x₀ such that for each s ∈ N, the set {x(s)| x is a solution of dx/dt = A(t) x} is equal to Rⁿ.

Proof. Let {c⁽¹⁾, ... , c⁽ⁿ⁾} be a basis of the vector space Rⁿ. By what had been established already, we can find a neighbourhood N₁ of s₀ and functions x⁽ⁱ⁾(t) such that dx⁽ⁱ⁾/dt = A(t) x⁽ⁱ⁾, x⁽ⁱ⁾(s₀) = c⁽ⁱ⁾, i = 1, ... , n. Let x⁽ⁱ⁾(t) = (a_i1, ... , a_in). Then, if

D(1) = 1. Hence, by continuity of the determinant function, we can find a neighbourhood N of s₀, contained in N₁, such that D ≠ 0 on N. Clearly N satisfies the requirements of the lemma.

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We now come to the matter of combining solutions on overlapping neighbourhoods. Let N and P be two intervals with nonempty intersection on which the solutions to the system are defined and satisfy the property stated in the lemma. Let r ∈ N ∩ P. Let {x⁽ⁱ⁾(t)}_{1 ≤ i ≤ n} and {y^(j)(t)}_{1 ≤ j ≤ n} be linearly independent solutions of dx/dt = A(t) x on N and P respectively. If

x⁽ⁱ⁾(t) = ∑_1≤j≤na_ij(t) c^(j)  and  y⁽ⁱ⁾(t) = ∑_1≤j≤nb_ij(t) c^(j)

we let

G(t) = [a_ij(t)]_1≤i,j≤n  and  H(t) = [b_ij(t)]_1≤i,j≤n.

Clearly G(r) = B H(r) for some nonsingular matrix B = [β_ij]_1≤i,j≤n. Extend the domain of x⁽ⁱ⁾(t), 1 ≤ i ≤ m to the whole of N ∪ P by setting x⁽ⁱ⁾(t) = ∑_jβ_ij y^(j)(t) on P. This gives ua a set of on the union which are linearly independent at each point.

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