If \(A\) is an \(n \times n\) matrix over the field \(F\) with characteristic polynomial \[ f = (x - c_1)^{d_1} \cdots (x - c_k)^{d_k} \] what is the trace of \(A\)?
Solution
The trace is \[ \boxed{\operatorname{tr}(A)=d_1c_1+\cdots+d_kc_k.} \]
Indeed, over a splitting field the Jordan form of \(A\) has exactly \(d_i\) diagonal entries equal to \(c_i\) for each \(i\). Similarity preserves trace, and the trace of a triangular matrix is the sum of its diagonal entries [@HK71]. Therefore \[ \operatorname{tr}(A)=\sum_{i=1}^k d_i c_i. \] Equivalently, this is the negative of the coefficient of \(x^{n-1}\) in the monic characteristic polynomial.