Cfrgtkky

Let $A$ be a $ntimes n$ matrix such that the characteristic polynomial of $A$ is $$P(lambda)=lambda^n+a_{n-1}lambda^{n-1}+...+a_1lambda+a_0$$

Now consider the nth order differential equation $$frac{d^nx}{dt^n}+a_{n-1}frac{d^{n-1}x}{dt^{n-1}}+...+a_1frac{dx}{dt}+a_0x=0$$ and let $f_0(t),: f_1(t),: ...,: f_{n-1}(t)$ be the unique solutions satisfying the inital conditions $$f_j^l(0)= begin{cases} 1 & j=l \ 0 & j neq lend{cases}$$ for $0 leq j,: lleq n-1$

Show that $e^{tA} = f_0(t)I+f_1(t)A+f_2(t)A^2+...+f_{n-1}(t)A^{n-1}$

I know that $e^{tA}$ is the function $M(t)$ such that $frac{dM}{dt}=AM$ and $M(0) = I$. So all I need to do is show that the right hand side of the equation satisfies these two conditions. The second one is easy enough to prove; however, the first one presents some trouble. My plan is to show that $frac{d^n}{dt^n}f_j(t) = A^nf_j(t)$ for every unique solution $f_j$, and based on the definitions of $A$ and the solutions to the differential equations, I suspect that Cayley-Hamilton theorem could be used here. However, I haven't been able to engineer it in a way to prove the above statement.

In the space of matrices $c(A)=c_0I+c_1A+...+c_{n-1}A^{n-1}$, the multiplication with $A$ results in
$$
A,c(A)=c_0A+c_1A^2+...+c_{n-2}A^{n-1}-c_{n-1}(a_0I+a_1A+...+a_{n-1}A^{n-1})
\
=-a_0c_{n-1}I+(c_0-a_1c_{n-1})A+...+(c_{n-2}-a_{n-1}c_{n-1})A^{n-1}
$$
This proves that the algebra generated by $A$ is spanned by the powers $I,A,...,A^{n-1}$. This means that there are function coefficients $f_0(t),...,f_{n-1}(t)$ with $e^{tA}=f_0(t)I+f_1(t)A+...+f_{n-1}(t)A^{n-1}$. From the differential equation $(e^{tA})'=Ae^{tA}$ we get thus
begin{align}
f_0'(t)&=-a_0f_{n-1}(t)\
f_1'(t)&=f_0(t)-a_1f_{n-1}(t)\
&~~vdots\
f_{n-2}'(t)&=f_{n-3}(t)-a_{n-2}f_{n-1}(t)\
f_{n-1}'(t)&=f_{n-2}(t)-a_{n-1}f_{n-1}(t)
end{align}
Inserting backwards one finds
$$
0=f_{n-1}^{(n)}(t)+a_{n-1}f_{n-1}^{(n-1)}(t)+...+a_1f_{n-1}'(t)+a_0f_{n-1}(t)
$$
with $0=f_{n-1}(0)=f_{n-1}'(0)=f_{n-1}^{(n-2)}(0)$ and $f_{n-1}^{(n-1)}(0)=1$.

Then conclude on the properties of $f_{n-2}(t)=f_{n-1}'(t)+a_{n-1}f_{n-1}(t)$ which is a solution of the same differential equation with $f_{n-2}^{(k)}(0)=0$ for $k<n-2$, $f_{n-2}^{(n-2)}(0)=1$ and $f_{n-2}^{(n-1)}(0)=f_{n-1}^{(n)}(0)+a_{n-1}=0$ and so on.

In other terms, assume that $P$ is not only the characteristic, but also the minimal polynomial of $A$. Set $$Q(x,y)=frac{P(x)-P(y)}{x-y},$$ $Q$ is a polynomial, so that $$P(D)I-P(A)=Q(D,A)(DI-A)$$ for the differentiation operator $D=frac{d}{dt}$ and identity matrix $I$. As $P(A)=0$ per Cayley-Hamilton, we get that any solution of $Dx=Ax$ also satisfies $P(D)x=0$.

Thus if $M(t)=sum_{k=0}^{n-1} f_k(t)A^k$ is a matrix solution, we get $sum_{k=0}^{n-1}P(D)f_k(t),A^k=0$, and as these powers of $A$ are linearly independent, $$P(D)f_k(t)=0$$ separately.

$M(0)=I$ implies $f_k(0)=delta_{0,k}$, $M^{(j)}(0)=A^j$ similarly implies $f_k^{(j)}(0)=delta_{j,k}$.

Now argue that $Q(D,A)$ is invertible on the solution space of the ODE system $Dx-Ax$ by the minimality of $P$ to find that a so constructed function $M$ reversely also satisfies $DM=AM$. Then use continuation from a dense set to also include matrices $A$ without the minimality of the characteristic polynomial.

Hint (but not a full answer): Using the definition of the matrix exponential, we write
$$
e^{tA} = sum_{k=0}^infty frac{1}{k!}(tA)^k .
$$
Then, using the Cayley-Hamilton theorem, we have $P(A) = 0$. Hence, all powers of $A$ with exponent larger or equal to $n$ can be expressed recursively as linear functions of $I, dots, A^{n-1}$:
$$
begin{aligned}
A^n &= -a_{n-1}A^{n-1} - dots - a_1 A - a_0 I \
A^{n+1} &= -a_{n-1}A^{n} - dots - a_1 A^2 - a_0 A \
&;; vdots
end{aligned}
$$
which we may be written
$$
A^k = sum_{i=0}^{n-1} b_i^{(k)} A^i , qquad kgeq n.
$$
Finally,
$$
begin{aligned}
e^{tA} &= sum_{k=0}^{n-1} frac{1}{k!}(tA)^k + sum_{k=n}^infty frac{1}{k!}(tA)^k \
&= sum_{k=0}^{n-1} frac{1}{k!}(tA)^k + sum_{k=n}^{infty} frac{1}{k!}t^k sum_{i=0}^{n-1} b_i^{(k)} A^i \
&= sum_{k=0}^{n-1} left( frac{1}{k!}t^k + sum_{p=n}^{infty} frac{1}{p!}t^p b_k^{(p)} right) A^k
end{aligned}
$$
which is of the expected form.