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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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Why, is that the time your homework is due? –  Gerry Myerson May 8 '12 at 12:33
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It was the exam time. This was a problem in the practice exam. –  Xiaowen Li May 21 '12 at 5:33
    
Good. I hope you did well. –  Gerry Myerson May 21 '12 at 5:40

1 Answer 1

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! –  Xiaowen Li May 8 '12 at 11:17

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