Compute the exponential of a matrix

Let D = \begin{bmatrix} 1 & 2 & -4 & 4 \\ 2 & -1 & 4 & -8 \\ 1 & 0 & 1 & -2 \\ 0 & 1 & -2 & 3 \end{bmatrix}. Compute \mathsf{exp}(D).


Let P = \begin{bmatrix} 0 & 1 & 2 & 0 \\ 2 & 0 & -2 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}. Evidently, P^{-1}DP = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} is in Jordan canonical form.

Now \mathsf{exp}(P^{-1}DP) = \mathsf{exp} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \oplus \mathsf{exp} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, using this previous exercise. By this previous exercise, we have \mathsf{exp} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} e & e \\ 0 & e \end{bmatrix}. So P^{-1}\mathsf{exp}(D)P = e P^{-1}DP, and we have \mathsf{exp}(D) = eD.

(WolframAlpha agrees.)

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