# Matrix exponentials

$$A = $\left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right)$$$

eigenvalues are $$\lambda_{1} = -1, \ \lambda_{2} = \lambda_{3} = 1$$

$$(A-\lambda_{1}I)u^{(1)} = 0 \ => \ u^{(1)} = $\left( \begin{array}{c} -1 \\ 0 \\ 1 \end{array} \right)$$$

$$(A-\lambda_{2}I)u^{(2)} = 0 \ => u^{(1)} = $\left( \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right)$$$

$$(A -\lambda_{3}I)u^{(3)} = u^{(2)} \ => \ \left( \begin{array}{ccc|c} 1&0&-1&-1\\ 0&0 &0&1\\ 0&0&0&0 \end{array} \right)$$

since we cannot have 0 = 1, we can say that there is only one eigenvector for

$$\lambda = 1$$

which means that the Jordan form will be

$$$\left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right)$$$

Am I correct?

Now I need to find exp(Jt) and I'm not sure how.

If I only have 2 eigenvectors, how can I find the fund. matrix T such that

$$e^{At} = Te^{Jt}T^{-1}$$?

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HallsofIvy
Homework Helper
In general, to find such a function of other than a "number", use the Taylor's series

eA= I+ A+ (1/2)A2+ (1/3!)A3+ ...

If A happens to be diagonal, that's easy: An is just the diagonal matrix with exponentials of the diagonal elements of A on its diagonal.

For Jordan form, its a little more complicated. Try calculating J2, J3, ... for this particular J yourself and see if you spot a pattern.

I noticed that A2k = I and A(2k+1)'s = A, but I don't know how to get eA from that.

Hurkyl
Staff Emeritus
Gold Member
I noticed that A2k = I and A(2k+1)'s = A, but I don't know how to get eA from that.
How exactly are you having trouble summing the series?

I'm just an idiot. I didn't realize that the Taylor expansions' limits came out to be cosh and sinh. I think I finished this problem. Thanks for the help so far.

Hurkyl
Staff Emeritus
Gold Member
For the record, there's another method that can be useful. Note that you can separate that Jordan block into

$$\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] + \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]$$

....

Ah, I see, and the second matrix is nilpotent.

But still, if I got that, how would I compute

$$e^{At} = Te^{Jt}T^{-1}$$

Not sure what T would be in this case.

Hurkyl
Staff Emeritus