In this previous question it came up that there's a pretty explicit formula for the exponential of a triangular matrix.

In the below, let $*$ denote convolution and define

$$ G_{\lambda_i}(t) = \begin{cases} \exp(\lambda_i t) & t \geq 0 \\ 0 & \text{otherwise.} \end{cases} $$

Then, given a lower triangular matrix

$$ L = \begin{pmatrix} \lambda_1 & & & \\ l_{21} & \lambda_2 & & \\ \vdots& & \ddots & \\ l_{n1}&l_{n2} &\dots &\lambda_n \end{pmatrix} $$

for $t \geq 0$, the $(a,b)$ entry of the matrix exponential is

$$ \left(\exp(tL)\right)_{ab} = \sum_{a = i_1 > i_2 > \dots > i_k = b} l_{i_1i_2}\cdots l_{i_{k-1}i_k} (G_{\lambda_{i_1}} * \dots * G_{\lambda_{i_k}})(t). $$

For example,

$$ \left(\exp(tL)\right)_{31} = l_{31} (G_{\lambda_3} * G_{\lambda_1})(t) + l_{32} l_{21} (G_{\lambda_3} * G_{\lambda_2} * G_{\lambda_1})(t). $$

This is fairly straightforward to see by induction. We're building up a solution to a differential equation $d/dt \exp(tA) = A \exp(tA)$. At each induction step there is an inhomogenous linear equation for the new entries, whose homogenous part has Green's function $G_{\lambda_n}$, so the inhomogenous solution is a combination of convolutions of the previous bits with that.

I suspect there's also some combinatorial interpretation and argument. Probably something in terms of Laplace transforms too.

This is a bit different from the way explicit formulas for exponentials of matrices are usually discussed:

  • Usually, we reduce things all the way to Jordan blocks rather than merely triangular matrices.
  • Usually, we decompose $G_{\lambda} * G_{\mu}$ into a linear combination $(\lambda - \mu)^{-1} G_\lambda - (\lambda - \mu)^{-1} G_\mu$, and handle the $\lambda = \mu$ case separately: $(G_\lambda * G_\lambda)(t) = t G_\lambda(t).$

(Avoiding both of these was useful for the previous question linked above; the goal was to find uniform bounds, and both the JNF and the decomposed-into-linear-combinations expressions are sensitive to small changes in the matrix entries.)

This direct formulation in terms of convolutions of Green's functions feels like a "textbook" result from some textbook somewhere, but I'm not familiar enough with the topic to know where.

Does anyone have a reference in the literature that states more or less this?



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