Linear Transformation and Diagonalization Problem

Stapler2000
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Nevermind -- Polygons and Polywags.
 
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matrix in standard basis looks good, but i would check your characteristic polynomial
 
i got x^2 - x + 6 = 0
 
however as you mention it is correct that in the case T has complex eigenvalues it is not diagonalisable over the reals
 
Agreed -- it is x^2-x+6.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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