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# Is this the right way to do this?

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Define the system of equations;

$$x_{n+1} = x_n - y_n$$

$$y_{n+1} = 2x_n + 4y_n$$

so if we take $$A = \begin{pmatrix} 1 & -1\\ 2 & 4 \end{pmatrix}$$

and define

$$\begin{pmatrix} x_{n+1} \\ y_{n+1} \\ \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 2 & 4 \\ \end{pmatrix}\begin{pmatrix} x_n \\ y_n \\ \end{pmatrix}$$

Then given that the eigenvalues and a corresponding eigenvector to the eigenvalue can be defined as

$$\lambda_1 = 3$$

$$\mu_1 = \begin{pmatrix} 1 \\ -2 \\ \end{pmatrix}$$

$$\lambda_2 = 2$$

$$\mu_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$$

Is it then true that the general solution can be defined as

$$x_n = 3^nc_1 + 2^nc_2$$

$$y_n = -2*3^nc_1 - 2^nc_2$$

Reinout

reinout-g  Jun 2, 2014

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Hi Reinout

Yes, agree with your result, with

c1 = -x0 - y0  and  c2 = 2x0 + y0 .

Bertie  Jun 2, 2014
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Perhaps I've misunderstood, but if c1 and c2 are initial values (n=0) for x and y then (for n = 1):

These don't look the same to me!

Alan  Jun 2, 2014
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Hey Alan,

you're right, but they are not initial values.

They are constants which can be calculated for given initial values for x and y

reinout-g  Jun 2, 2014
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