# Characteristic polynomial coefficients

1. Sep 29, 2011

### 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

2. Sep 29, 2011

### 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.