I have been working on a problem where I need to Taylor expand an expression of the form $log det(I-A)$ in terms of traces of the matrices $A^m$ for $m in mathbb N$, where $A$ is a general $n times n$ matrix.

I did notice that if the eigenvalues of $A$ are $lambda_1,

cdots , lambda_n$ then those of $I-A$ are exactly $1 – lambda_1, cdots , 1- lambda_n$, so we may write

$$log det(I-A) = sum_{i=1}^n log (1 – lambda_i) = sum_{i=1}^n sum_{m=1}^infty frac{(-1)^m lambda_i^m}{m} = sum_{m=1}^infty frac{(-1)^m}{m} sum_{i=1}^n lambda_i^m hspace{10mm} cdots (1)$$

At this point, I noticed that if $A$ were diagonalizable then I could say that $P^{-1}AP = diag{lambda_1, cdots , lambda_n}$ (the diagonal matrix with entries $lambda_1, cdots , lambda_n$ along the principal diagonal), so for every $m geq 1$, I could write $P^{-1}A^mP = diag{lambda_1^m, cdots , lambda_n^m }$ and $tr(A^m) = tr(P^{-1}A^mP) = sum_{i=1}^n lambda_i^m$ and (1) would then give us

$$log det(I-A) = sum_{m=1}^infty frac{(-1)^m}{m} tr(A^m)$$

which is what I want. But I couldn’t get around the case when $A$ was non-diagonalizable. I was wondering what happens in that case. Can we still give the same (or maybe similar) expansions? Would the Smith Normal Form, Rattional Canonical Form etc. be of any help?

**P.S.:** I didn’t find any standard reference containing the kind of expansion I wanted. I would appreciate if I would come to know what is the best thing one can say in the diagonalizable case, and/or if i were pointed out to some reference containing with these kinds of result(s).