- #1
TheFerruccio
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Homework Statement
Find the eigenbasis and diagonalize.
Homework Equations
[tex]\mathbf{A} = \left[ {\begin{array}{ccc}
5& \frac{8}{3} & \frac{-2}{3} \\
2 & \frac{2}{3}& \frac{4}{3} \\
-4 & \frac{-4}{3} & \frac{-8}{3}\\
\end{array} } \right][/tex]
The Attempt at a Solution
I find the characteristic equation by finding the determinant of [tex]\mathbf{A} - \lambda \mathbf{I}[/tex][tex]\left|\mathbf{A} - \lambda \mathbf{I}\right| = \left| {\begin{array}{ccc}
5 - \lambda & \frac{8}{3} & \frac{-2}{3} \\
2 & \frac{2}{3} - \lambda & \frac{4}{3} \\
-4 & \frac{-4}{3} & \frac{-8}{3} - \lambda\\
\end{array} } \right|[/tex] = 0
This gets me the cubic equation:
[tex]-\lambda^3 + 3\lambda^2 + 18\lambda = 0[/tex]
So, here's the question: Are there any nice, fast ways to get the roots of the cubic equation? Furthermore, is there any faster way to find the characteristic polynomial that doesn't include such a high probability of arithmetic error when doing it by hand?
All of this is done by hand, no calculator, pencil and paper.
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