Derivative of matrix exponential of linear combination

Question

Let's say $t$ is a real parameter and $\textbf{A}$ is $n \times n$ matrix. I know that

$$\frac{d}{dt} \exp(t\textbf{A}) = \textbf{A} \exp(t\textbf{A})$$

But what if there are multiple parameters $t_1, ..., t_n$ and multiple matrices $\textbf{A}_1, \dots , \textbf{A}_n$ that don't commute, does the same relation, namely

$$\frac{\partial}{\partial t_k}\exp \left(\sum_{i=1}^n t_i \textbf{A}_i \right) = \textbf{A}_k \exp \left(\sum_{i=1}^n t_i \textbf{A}_i\right)$$

still hold? If not, is there a closed form for the derivative?

Answer 1

PARTIAL ANSWER.

If there is a general formula for $$ \partial_t e^{tX+sY}, \qquad \partial_s e^{tX+sY}, $$ then it surely descends from the Baker-Campbell-Hausdorff formula, which is known to be rather complicated. So, I will only consider a simpler special case.

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In the special case of $X$ and $Y$ commuting with the commutators, that is $$\tag{1}[X, [X, Y]]=[Y, [X, Y]]=0,$$ the Baker-Campbell-Hausdorff formula simplifies to $$\tag{2} e^{tX+sY+\frac{ts}{2}[X, Y]}=e^{tX}e^{sY}, $$ and since, by (1), $tX+sY$ commutes with $[X, Y]$,
$$ e^{tX+sY}=e^{tX}e^{sY}e^{-\frac{ts}{2}[X, Y]}, $$ from which we infer $$\begin{split} \partial_t e^{tX+sY} &= Xe^{tX}e^{sY}e^{-\frac{ts}{2}[X,Y]} - e^{tX}e^{sY}\left(\frac{s}{2}[X, Y]\right)e^{-\frac{ts}{2}[X,Y]}\\ &=(X-\frac{s}{2}[X, Y])e^{tX+sY}, \end{split}$$ where we used (1), which implies that $[X,Y]$ commutes with $e^{tX}$ and with $e^{sY}$.

Reversing the roles of $X$ and $Y$, and using that $[Y, X]=-[X, Y]$, we have $$ \partial_s e^{tX+sY} = (Y+\frac{t}{2}[X, Y])e^{tX+sY}.$$

In particular, we see that in general it is not true that $$\tag{!!} \partial_t e^{tX+sY} = X e^{tX+sY}.$$

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SOME WORDS ON THE GENERAL CASE.

If we drop the assumption (1), then the Baker-Campbell-Hausdorff formula becomes much more complicated; $$\tag{3} e^{tX+sY+\frac{ts}{2}[X, Y]+\frac1{12}\left( t^2s[X,[X,Y]] - ts^2[Y, [X,Y]]\right)+\ldots} =e^{tX}e^{sY}.$$ I don't know how to pass from (3) to a formula of the form $$ e^{tX+sY}=e^{tX}e^{sY}e^{\Phi(t, s)},$$ which is what we need for the above computation.

Answer 2

As for analytical functions in one dimension, one can define a matrix function for square matrices $\mathbf{X}$ by an infinite series: $$ f(\mathbf{X}) = \sum_{n=0}^\infty c_n \mathbf{X}^n$$ assuming the limit exists and is finite. If the coefficients $c_n$ fulfils $\sum c_n x^n <\infty$, then one can prove that the above series exists and is finite

_{source: The matrix cookbook}

So we know that $\exp(t\mathbf{A})$:

$$ \exp(t\mathbf{A}) = \sum_{n=0}^\infty \frac{t^n \mathbf{A}^n}{n!} $$

and thus

$$ \frac{\textrm{d}}{\textrm{d}t}\exp(t\mathbf{A}) = \sum_{n=1}^\infty \frac{n t^{n-1} \mathbf{A}^n}{n!} = \sum_{n=0}^\infty \frac{t^{n} \mathbf{A}^{n+1}}{n!}=\textbf{A}\,\exp(t\mathbf{A}) = \exp(t\mathbf{A})\,\textbf{A} $$

The OP is interested in

$$ \frac{\partial}{\partial t_p}\exp\left\{\sum_{i=1}^nt_i\mathbf{A}_i\right\}. $$

This is not as straightforward as it seems. To explain this you need to understand that:

$$ \exp(\mathbf{A} + \mathbf{B}) = \exp(t\mathbf{A})\,\exp(t\mathbf{B}),\quad\textrm{if}\quad \mathbf{AB}=\mathbf{BA} $$

If $\mathbf{A}$ and $\mathbf{B}$ do not commute, this is not the case. You can see this from the simple expansion:

$$ \exp(\mathbf{A} + \mathbf{B}) = \sum_{n=0}^\infty \frac{t^n (\mathbf{A}+\mathbf{B})^n}{n!}$$

Only if $\mathbf{A}$ and $\mathbf{B}$ commute, you can use the binomial expansion, in the other case it becomes really messy. To give an example, the second term becomes:

$$\begin{align} \exp(\mathbf{A} + \mathbf{B}) =& \mathbf{I} + \mathbf{A}+\mathbf{B} \\ & + \frac{1}{2}\left(\mathbf{A}^2 + \mathbf{AB} + \mathbf{BA} + \mathbf{B}^2\right) \\ & + \frac{1}{3!}\left(\mathbf{A}^3 + \mathbf{A}^2\mathbf{B} + \mathbf{A}\mathbf{B}\mathbf{A} +\mathbf{B}\mathbf{A}^2 + \mathbf{B}^2\mathbf{A} + \mathbf{B}\mathbf{A}\mathbf{B} +\mathbf{A}\mathbf{B}^2+ \mathbf{B}^3\right)\\ &+\ldots \end{align}$$

So, in the OP, if $\mathbf{A}_p\mathbf{A}_q = \mathbf{A}_q\mathbf{A}_p$ for all combinations of $p$ and $q$, then

$$\frac{\partial}{\partial t_p}\exp\left\{\sum_{i=1}^nt_i\mathbf{A}_i\right\} = \textbf{A}_p\,\exp\left\{\sum_{i=1}^nt_i\mathbf{A}_i\right\} = \exp\left\{\sum_{i=1}^nt_i\mathbf{A}_i\right\}\,\textbf{A}_p$$

If the matrices do not commute, i.e. $\mathbf{A}_p\mathbf{A}_q \ne \mathbf{A}_q\mathbf{A}_p$ for any combination of $p$ and $q$, it quickly blows up.

Answer 3

Let $$\eqalign{ t&=t_i,\quad A=A_i,\quad B=\sum_{k\ne i} t_kA_k \\ G(t) &= \frac{d}{dt}\exp(B+tA) \\ }$$ To evaluate $G(t)$ at $t=0$, use the block-triangular method. $$\eqalign{ \exp\left(\left[\matrix{B&A\\0&B}\right]\right) &= \left[\matrix{\exp(B)&G(0)\\0&\exp(B)}\right] \\ }$$ To evaluate $G$ at $t=s,\,$ shift the definition of $B\to (B+sA)$

Define block-analogs of the standard basis vectors $$e_1=\pmatrix{1\\0},\quad e_2=\pmatrix{0\\1}$$ by replacing the {${0,1}$} elements with the $n\times n\,$ {zero, identity} matrices $$E_1=\pmatrix{I\\0},\quad E_2=\pmatrix{0\\I} \in{\mathbb R}^{2n\times n}$$ Then would this qualify as a closed-form solution? $$\eqalign{ G(s) &= E_1^T\,\exp\left(\left[\matrix{B+sA&A\\0&B+sA}\right]\right)\,E_2 \\ }$$