Putzer's Algorithm

[meks0899]

i came across this problem in my class notes. I'm curious about the problem. Any help would be appreciated.

Thanks

Use Putzer's Algorithm to find e^At given that
A = [1 -1]
[5 -1]

 

[galactus]

Wow. Cool problem. Putzer's algorithm is not that old, so it is not that well known.

But, having researched matrix exponentials, I think I may be able to help a wee bit.

By Putzer's algorithm, $\displaystyle e^{At}=p_{1}(t)M_{0}+p_{2}(t)M_{1}$

Where $\displaystyle M_{0}=I, \;\ M_{1}=\begin{bmatrix}a-{\lambda}_{1}&b\\c&d-{\lambda}_{1}\end{bmatrix}$

$\displaystyle p_{1}(t)=e^{{\lambda}_{1}t}, \;\ p_{2}(t)=\frac{1}{{\lambda}_{1}-{\lambda}_{2}}\left(e^{{\lambda}_{1}t}-e^{{\lambda}_{2}t}\right)$

Of course, I assume you know that the 2 lambdas represent the eigenvalues of matrix A.

The characteristic polynomial of said matrix is $\displaystyle {\lambda}^{2}+4$. Which means the eigenvalues are $\displaystyle {\lambda}_{1}=2i, \;\ {\lambda}_{2}=-2i$

Now, proceeding. I am not going to go into every derivation and proof of why and/or how this works. You can probably find all that somewhere if you need it.

Therefore, from all of the above:

$\displaystyle e^{At}=\begin{bmatrix}e^{{\lambda}_{1}t}+\frac{a-{\lambda}_{1}}{{\lambda}_{1}-{\lambda}_{2}}\left(e^{{\lambda}_{1}t}-e^{{\lambda}_{2}t}\right)&\frac{b}{{\lambda}_{1}-{\lambda}_{2}}\left(e^{{\lambda}_{1}t}-e^{{\lambda}_{2}t}\right)\\ \frac{c}{{\lambda}_{1}-{\lambda}_{2}}\left(e^{{\lambda}_{1}t}-e^{{\lambda}_{2}t}\right)& e^{{\lambda}_{1}t}+\frac{d-{\lambda}_{1}}{{\lambda}_{1}-{\lambda}_{2}}\left(e^{{\lambda}_{1}t}-e^{{\lambda}_{2}t}\right)\end{bmatrix}$

Plug in a,b,c,d and the eigenvalues.

The matrix exponential is used to solve DE's. So, I suppose this is all that is needed because I see no initial values.