Properties of the derivative of a matrix power series

Let G(x) be a formal power series on \mathbb{C} having an infinite radius of convergence and fix an n \times n matrix A over \mathbb{C}. The mapping t \mapsto G(At) carries a complex number t to a complex matrix G(At); we can think of this as the ‘direct sum’ of n \times n different functions on \mathbb{C}, one for each entry of G(At).

We now define the derivative of G(At) with respect to t to be a mapping \mathbb{C} \rightarrow \mathsf{Mat}_n(\mathbb{C}) as follows: \left[\frac{d}{dt} G(At)\right]_{i,j} = \frac{d}{dt} \left[ G(At)_{i,j}\right]. In other words, thinking of G(At) as a matrix of functions, \frac{d}{dt} G(At) is the matrix whose entries are the derivatives of the corresponding entries of G(At).

We will use the limit definition of derivative (that is, \frac{d}{dt} f(t) = \mathsf{lim}_{h \rightarrow 0} \dfrac{f(t+h) - f(t)}{h}, where it doesnt matter how h approaches 0 in \mathbb{C}) and will assume that all derivatives exist everywhere.

Prove the following properties of derivatives:

  1. If G(x) = \sum_{k \in \mathbb{N}} \alpha_kx^k, then \frac{d}{dt} G(At) = A \sum_{k \in \mathbb{N}} (k+1)\alpha_{k+1}(At)^k.
  2. If V is an n \times 1 matrix with constant entries (i.e. not dependent on t) then \frac{d}{dt} (G(At)V) = \left( \frac{d}{dt} G(At) \right) V.

[My usual disclaimer about analysis applies here: as soon as I see words like ‘limit’ and ‘continuous’ I become even more confused than usual. Read the following with a healthy dose of skepticism, and please point out any errors.]

Note the following.

\left[ \dfrac{d}{dt} G(At) \right]_{i,j}  =  \dfrac{d}{dt} G(At)_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \dfrac{G(A(t+h)_{i,j}) - G(At)_{i,j}}{h}
 =  \mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{G(A(t+h)) - G(At)}{h} \right]_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_k \alpha_k(A(t+h))^k - \sum_k \alpha_k(At)^k}{h} \right]_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_k \alpha_k A^k ((t+h)^k - t^k)}{h} \right]_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_{k > 0} \alpha_k A^k \left( \sum_{m=0}^k {k \choose m} t^m h^{k-m} - t^k \right)}{h} \right]_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_{k > 0} \alpha_k A^k \left( \sum_{m=0}^{k-1} {k \choose m} t^m h^{k-m} \right)}{h} \right]_{i,j}
 =  \mathsf{lim}_{h \rightarrow 0} \displaystyle\sum_{k > 0} \alpha_k A^k \sum_{m=0}^{k-1} {k \choose m} t^m h^{k-1-m} (Now we can substitute h = 0.)
 =  \displaystyle\sum_{k > 0} \alpha_k A^k {k \choose {k-1}} t^{k-1} (All terms but m=k-1 vanish.)
 =  \displaystyle\sum_{k > 0} k \alpha_k A^k t^{k-1}
 =  \displaystyle\sum_k (k+1) \alpha_{k+1}A^{k+1}t^k
 =  A \sum_k (k+1)\alpha_{k+1}(At)^k

As desired.

Now say V = [v_i] and G(At) = [c_{i,j}(t)]; we then have the following.

\dfrac{d}{dt} \left( G(At)V \right)  =  \dfrac{d}{dt} \left( [c_{i,j}(t)][v_{i,j}] \right)
 =  \dfrac{d}{dt} [\sum_k c_{i,k}(t)v_k]
 =  [\frac{d}{dt} \sum_k c_{i,k}(t)v_k]
 =  [\sum_k (\frac{d}{dt} c_{i,k}(t)) v_k]
 =  [\frac{d}{dt} c_{i,j}(t)][v_i]
 =  \left(\frac{d}{dt} G(At) \right)V

As desired.

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