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We are given a non-linear system:

$4x_1 − x_2 + x_3 = x_1x_4,$

$−x_1 + 3x_2 − 2x_3 = x_2x_4$

$x_1 − 2x_2 + 3x_3 = x_3x_4$

$x_1^2 + x_2^2 + x_3^2 = 1$

And the question asks: Show how to solve the nonlinear system via the computation of all the eigenvalues and eigenvectors of a 3 × 3 matrix

It is likely the question asks for a numerical method.

If you can provide some hint beofore 8am, PST, I would really appreciate it:)

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    Good. I hope you did well.2012-05-21

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Maybe it helps if you express everything in matrix-vector format, like so:

$\begin{pmatrix}4&-1&1\\-1&3&-2\\1&-2&3\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=x_4\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$

The last equation is likely mistaken, and is supposed to be $x_1^2+x_2^2+x_3^2=1$ (a normalization condition). Can you take it from here?

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    Yes this is also exactly what I came up with. Thanks for your help!2012-05-08