characteristic polynomial coefficients

JamesGoh

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

mathwonk

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.