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Characteristic polynomial coefficients

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JamesGoh
#1
Sep29-11, 10:38 PM
P: 140
For any characteristic polynomial determined from A - eI (where A is a nxn matrix, e is an eigenvalue and I is the identity matrix),

is it a rule that the coefficient associated with the char. polynomail term of highest degree must be positive ?

My tutor made a theory that if the characteristic polynomial's coefficient of highest power is negative, then divide it through by -1 to make it positive.

Im not sure why it has to be positive, so I'd really like some clarification

thanks
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mathwonk
#2
Sep29-11, 11:27 PM
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there is no rule on this. there is an annihilator ideal and the characteristic polynomial is any generator, so the first coefficient could almost be anything. i usually like it to be positive. but if you define it to be a certain determinant t could be either. this is not at all important for applications.


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