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I have a $4\times4$ matrix $A$, which is composed of two $2 \times 2$ Jordan blocks.

The matrix in question is the matrix associated with a particular system of differential equations.

so, $$e^{Ax} = \sum_{k = 0}^{\infty}\frac{A^kx^k}{k!} = \sum_{k = 0}^{\infty}\frac{ \begin{bmatrix} L & 0 \\ 0 & K \end{bmatrix} ^kx^k}{k!} = \sum_{k = 0}^{\infty}\frac{ \begin{bmatrix} L^k & 0 \\ 0 & K^k \end{bmatrix} x^k}{k!}$$

I feel uncomfortable manipulating block matrices(block matrix-vector multiplication).

what would be the next step?

I would like to have something along the lines of $e^{Kx}e^{Lx}$ I could try:

$$e^{A} = \sum_{k = 0}^{\infty}\frac{A^k}{k!} = \sum_{k = 0}^{\infty}\frac{ \begin{bmatrix} L & 0 \\ 0 & K \end{bmatrix} ^k}{k!} = \sum_{k = 0}^{\infty}\frac{ \begin{bmatrix} L^k & 0 \\ 0 & K^k \end{bmatrix} }{k!}= \begin{bmatrix} e^L & 0 \\ 0 & e^K \end{bmatrix}$$

Then $e^{Ax} = [e^A]^x =[\begin{bmatrix} e^L & 0 \\ 0 & e^K \end{bmatrix} ]^x = \color{red}{\begin{bmatrix} e^{Lx} & 0 \\ 0 & e^{Kx} \end{bmatrix}}$ Would this logic work out?

  • Yes, the exponential can be taken clockwise. See also Why does the $n$-th power of a Jordan matrix involve the binomial coefficient? for the explicit form of the matrix exponential of a Jordan block. – user1551 Mar 23 '16 at 10:41
  • Thanks, I worked that part out by decomposing A into a diagonal matrix(containing eigenvectors) and the off diagonal parts, It was just bridging the gap. Let me edit my post. –  Mar 23 '16 at 10:52
  • Is that statement, i.e $e^{Ax} = \text{ red matrix}$ legitimate? –  Mar 23 '16 at 10:56
  • The red one is OK, but the two in the middle are unnecessary complications, because you have yet to define $(e^A)^x$. You have shown in the displayed line that $e^A=\pmatrix{e^L\ & e^K}$ whenever $A$ is in the form of $\pmatrix{L\ &K}$. So, in particular, if you let $xA$ takes the role of $A$ (I suppose that $x$ is a scalar), then you immediately get $e^{xA}=\pmatrix{e^{xL}\ &e^{xK}}$. – user1551 Mar 23 '16 at 11:25
  • Oh I see. Thank you :) –  Mar 23 '16 at 13:40

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