Proposition 5

The determinant of the matrix $C(t, t_0)$ is given by the formula

(15)

Indeed, if one puts $\delta(t) = \det \left( C(t, t_0) \right)$ one has, by the formula for the derivative of a determinant (I, p. 8, formula (3))

that is, by the differential equation (5) of IV, p. 179 satisfied by $C(t, t_0)$,

Since $\delta(t_0) = 1$ the formula (15) follows from the expression (12) (IV, p. 183) for the integral of a scalar linear equation.

Specifying $n$ linearly independent integrals of (4) determines all the integrals of this equation, as we have just seen. We shall now show that for $1 \leq p \leq n$ the knowledge of $p$ linearly independent integrals $\mathbf{u}_j$ ($1 \leq j \leq p$) of equation (4) reduces the integration of this equation to that of a homogeneous linear system of $n - p$ scalar equations. Suppose that on an interval $K \subset J$ there are $n - p$ maps

of $K$ into $E$, which are primitives of regulated functions on $K$, and such that, for every $t \in K$, the $n$ vectors $\mathbf{u}_j(t)$ ($1 \leq j \leq n$) form a basis of $E$.

For every point $t_1 \in J$ there always exists an interval $K$, a neighbourhood of $t_1$ in $J$, on which there are defined $n - p$ functions $\mathbf{u}_{p+k}$ ($1 \leq k \leq n - p$) having the preceding properties. For, let $(\mathbf{e}_i)_{1 \leq i \leq n}$ be a basis of $E$; there exist $n - p$ vectors of this basis which form with the $\mathbf{u}_j(t_1)$ ($1 \leq j \leq p$) a basis of $E$ (Alg., II, p. 292, th. 2); suppose for example that they are $\mathbf{e}_{p+1}, \ldots, \mathbf{e}_n$; since the determinant $\det(\mathbf{u}_1(t), \ldots, \mathbf{u}_p(t), \mathbf{e}_{p+1}, \ldots, \mathbf{e}_n)$ (with respect to the basis $(\mathbf{e}_i)$) is a continuous function of $t$ and does not vanish for $t = t_1$ there exists a neighbourhood $K$ of $t_1$ on which it does not vanish; one can then take $\mathbf{u}_{p+k}(t) = \mathbf{e}_{p+k}$ ($1 \leq k \leq n - p$) for $t \in K$.

There exists an invertible matrix $B(t)$ of order $n$, whose elements are primitives of regulated functions on $K$, such that $B(t).\mathbf{e}_j = \mathbf{u}_j(t)$ for $1 \leq j \leq n$. Let us put $x = B(t).y$; then $y$ satisfies the equation $\frac{dB}{dt}.y + B(t).\frac{dy}{dt} = A(t)B(t).y$, which can also be written

where $H(t) = (h_{jk}(t))$ is a matrix with regulated entries on $K$. By the definition of $B(t)$ this linear equation admits the $p$ constant vectors $\mathbf{e}_j$ ($1 \leq j \leq p$) as integrals; one concludes immediately that necessarily $h_{jk}(t) = 0$ for $1 \leq k \leq p$; thus the components $y_k$ of $y$ (with respect to the basis $(\mathbf{e}_i)$) with index $k \geq p + 1$ satisfy a homogeneous linear system of $n - p$ equations; once the solutions of this system are determined, the $dy_j/dt$ for indices $j \leq p$ are linear functions of the $y_k$ with $k \geq p + 1$, so are known, and the primitives of these functions will give the $y_j$ for indices $j \leq p$.

In particular, when one knows $n - 1$ linearly independent integrals of equation (4) of IV, p. 183, integrating this equation reduces to that of a single homogeneous scalar equation, and then the evaluation of $n$ primitives.