Not sure how to write a matrix in TeX, but suppose I have a canonical matrix of Jordan blocks along its diagonal. If I take the exponential of this matrix B with $e^{Bt}$, is the resulting matrix just the exponential of each of those blocks but within the same respective location in the resulting matrix?
Like if $B_{1}=\{\{a,-b\},\{b,a\}\}$ and $B_{2}=\{\{c,-d\},\{c,d\}\}$ in a $4x4$ matrix with all other entries being 0, how would I compute the result? There's two distinct real components for each pair of complex eigenvectors, so where do I put the $e^{a}$ and the $e^{c}$? Do they both go on the outside of the overall $4x4$ matrix or are they both multiplied by their respective blocks within the resulting $4x4$ matrix?
Hint For a direct sum $$\pmatrix{P\\&Q}$$ of square matrices, induction gives $$\pmatrix{P\\&Q}^k = \pmatrix{P^k\\&Q^k} ,$$ so substituting in the usual power series for $\exp$ gives \begin{align} \exp \pmatrix{P\\&Q} &= \pmatrix{I\\&I} + \pmatrix{P\\&Q} + \frac{1}{2} \pmatrix{P\\&Q}^2 + \cdots \\ &= \pmatrix{I\\&I} + \pmatrix{P\\&Q} + \frac{1}{2} \pmatrix{P^2\\&Q^2} + \cdots \\ &= \pmatrix{I + P + \frac{1}{2} P^2 + \cdots\\&I + Q + \frac{1}{2} Q^2 + \cdots} \\ &= \pmatrix{\exp P\\&\exp Q} \end{align}