Proposition 7

1' The map $X \mapsto e^X$ is a continuous map of $\mathscr{L}(E)$ into the group of automorphisms of E (invertible elements of* $\mathscr{L}(E)$).

2 The map $t \mapsto e^{Xt}$ of $\mathbf{R}$ into $\mathscr{L}(E)$ is differentiable, and

(24)

3 For any $X \in \mathscr{L}(E)$ one has

(25)

the right-hand side being absolutely and uniformly convergent on every bounded subset of $\mathscr{L}(E)$; in particular, $e^{It} = e^t I$ for $t \in \mathbf{R}$.

4) If X and Y commute then Y and $e^Y$ commute with $e^X$, and

(26)

The relation (24) follows from the expression (23) for $C(t, 0)$ and from the fact that this function is an integral of (20); by recursion on n one deduces from (24) that $t \mapsto e^{Xt}$ is indefinitely differentiable on $\mathbf{R}$ and that

By Taylor’s formula one thus can write

(27)

On the other hand, cor. 3 of IV, p. 181 shows that $\| e^{Xt} \| \leq \exp (\| X \| |t| )$. Thus the remainder $r_n(X) = X^{n+1} \int_0^1 \frac{(1-t)^n}{n!} e^{Xt} \, dt$ in formula (27) satisfies the inequality

whence one deduces the formula (25), the series on the right-hand side being absolutely and uniformly convergent on every bounded subset of $\mathscr{L}(E)$. For every pair of elements $X,\ T$ of $\mathscr{L}(E)$ one thus has

Now, one can write $(X + T)^n - X^n = \sum_{(V_i)} V_1 V_2 \ldots V_n$, where the sum is taken over the $2^n - 1$ sequences $(V_i)$ of elements of $\mathscr{L}(E)$ such that $V_i = X$ or $V_i = T$ for $1 \leq i \leq n$, and at least one of the $V_i$ is equal to $T$; the inequality

follows immediately, whence which establishes the continuity of the map $X \mapsto \exp X$.

Finally, if $X$ and $Y$ commute, then $Y$ commutes with $e^{Xt}$ (IV, p. 181, prop. 2), so

Since, on the other hand, $e^{Xt} e^{Yt}$ is equal to $I$ for $t = 0$, one has $e^{Xt} e^{Yt} = e^{(X+Y)t}$, whence formula (26). From this latter, one deduces in particular that for arbitrary real $s$ and $t$ one has and also that

On the contrary one notes that (26) need not remain valid if $X$ and $Y$ are no longer assumed to commute: indeed it would imply that $\exp X$ and $\exp Y$ always commute, which is not the case, as is shown by simple examples (IV, p. 204, exerc. 3).

Now let us assume that $E$ is a vector space of finite dimension over the field $\mathbf{C}$, and $A$ an endomorphism of $E$ (for its vector space structure over $\mathbf{C}$) which one can identify with its matrix with respect to a basis of $E$; then, for all $t \in \mathbf{R}$, $e^{At}$ is an automorphism of $E$ for the same structure (IV, p. 181, prop. 2). Let $r_k$ ($1 \leq k \leq q$) be the distinct roots (in $\mathbf{C}$) of the characteristic polynomial $\varphi(r) = \det(A - rI)$ of the endomorphism $A$ (the "characteristic roots" of $A$); if $n_k$ is the order of multiplicity of $r_k$ then $\sum_{k=1}^q n_k = n$. One knows that (Alg., VII, 31, n° 3) to each root $r_k$ there corresponds a subspace $E_k$ of $E$, of dimension $n_k$, such that $E_k$ is stable under $A$, and that $E$ is the direct sum of the $E_k$: $E_k$ can be defined as the subspace of vectors $\mathbf{x}$ such that

Let $\mathbf{a}$ be any vector in $E$; one can write $\mathbf{a} = \sum_{k=1}^q \mathbf{a}_k$, where $\mathbf{a}_k \in E_k$; the integral of the equation (19) of IV, p. 188, taking the value $\mathbf{a}$ at the point $t = 0$ is thus given by

But since $\mathbf{a}_k \in E_k$ one has

Thus every integral of the equation (19) of IV, p. 188, can be written as

where $\mathbf{p}_k(t)$ is a polynomial in $t$, with coefficients in the vector space $E_k$, and of degree $\leq n_k - 1$. In particular, if all the roots of the characteristic equation of $A$ are simple, the spaces $E_k$ ($1 \leq k \leq n$) are all of dimension 1 over the field $\mathbf{C}$, so there exist $n$ vectors $c_k$ such that the $n$ functions $e^{r_k t} c_k$ ($1 \leq k \leq n$) form a fundamental system of integrals of the equation (19) of IV, p. 188.

The characteristic roots of the endomorphism $A$ are also called characteristic roots of the linear equation (19) of IV, p. 188. One may observe that one obtains the characteristic equation of $A$ by writing that the function $c e^{r t}$ is an integral of (19) for a vector $c \neq 0$.

When one has determined the roots $r_k$ ($1 \leq k \leq q$) explicitly, and thus the order of multiplicity $n_k$ of $r_k$, in practice one obtains the integrals of (19) by writing that this equation is satisfied by the expression (32) of IV, p. 191, where $\mathbf{p}_k$ is an arbitrary polynomial of degree $\leq n_k - 1$, with coefficients in $E$; on identifying the coefficients of $e^{r_k t}$ (for $1 \leq k \leq q$) in the two sides of the equation so obtained one obtains linear equations for the coefficients of the polynomials $\mathbf{p}_k$: one establishes easily that these equations determine the terms of degree $> 0$ of $\mathbf{p}_k$ as functions of the constant term, and that the latter is a solution of the equation $(A - r_k I)^{n_k} \cdot \mathbf{x} = 0$, which defines the subspace $E_k$ (method of "undetermined coefficients").