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If I have a matrix

$$A = \begin{bmatrix}5 & 4 & -6\\-2 & -1 & 2\\2 & 0 & -3\end{bmatrix}$$

how do I solve $x'=Ax$ as a differential equation? My text book explains this in a rather confusing way and I am really not getting it - what if I just found the eigen values - would the solution be anything to do with that or could I just use some kind of gaussian elimination?

Any help would be much appreciated, many thanks. :)

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apart from those two answers, it is well explained on here –  Santosh Linkha Mar 20 '13 at 13:39
    
If you like an answer you can accept it –  Dominic Michaelis Mar 21 '13 at 21:03
    
sorry i have now! :) –  user67411 Mar 27 '13 at 12:13

2 Answers 2

up vote 3 down vote accepted

You need the Eigenvalues, the main idea is that $$\frac{d}{dt} e^{At}= A e^{At}$$ To compute $e^{At}$ you need the eigenvalues and their multiplicity, gaussian elimenation won't help you.

The Eigenvalues of your matrix are $$\sigma=\{ 1+2i,1-2i,1\}$$ and the eigenvectors are $$\begin{pmatrix} 2+i \\ - 1 \\ 1 \\ \end{pmatrix} \quad \begin{pmatrix} 2-i\\ -1\\ 1 \end{pmatrix} \quad \begin{pmatrix} 1\\ 0 \\ 1 \end{pmatrix}$$ So the system is an linear combination of $e^{\lambda_i t} v_i$.

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so is each answer $e^At$ where A is an eigen value? –  user67411 Mar 20 '13 at 13:38
    
no $A$ is the matrix, do you know jordan normal form? the solutions will be a combination of eigen and main ( haupt?) vectors –  Dominic Michaelis Mar 20 '13 at 13:41
    
im not sure i get it - jordan normal form hasnt been mentioned! –  user67411 Mar 20 '13 at 13:46
    
@user67411 added a bit –  Dominic Michaelis Mar 20 '13 at 14:04
    
ah okay great i get that now! why couldnt my book say it like that?! So is that it? Many thanks –  user67411 Mar 20 '13 at 14:07

The solution to your differential equation is $$x(t) = e^{At}$$ where $e^{A}$ is defined as $$e^A = \sum_{i=0}^{\infty} \frac{A^k}{k!}.$$ This infinite series can be evaluated by writing $A = PDP^{-1}$, the eigenvalue decomposition of A (assuming it exists), where $D$ is a diagonal matrix of eigenvalues of $A$. For an example, see http://www.millersville.edu/~bikenaga/linear-algebra/matrix-exponential/matrix-exponential.html.

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