I have to study/evaluate many determinants of the form
$$
 f\_M(J)=\det(J-M),
$$
where $M$ is fixed, and $J$ is a diagonal matrix (with
0/1 on the diagonal, if it helps.) In my problem
 $M$ is fixed, and $J$ varies. 
Any suggestions?



determinant of diagonal - fixed
Let $J$ be $\mathrm{diag}(a\_1,...,a\_n)$. Let $M\_{i\_1,...,i\_k}$ be the principal submatrix of $M$ with rows and colums number $\{i\_1,...,i\_k\}$ deleted. Then $\det(J+M)=\det(M+\mathrm{diag}(0,a\_2,...,a\_n))+a\_1\det(J\_1+M\_1)$ (expand along the first row). Thus $$\det(J+M)=\sum a\_{i\_1}\cdot\ldots\cdot a\_{i\_k} \det(M\_{i\_1,...,i\_k}).$$ The sum is taken over all subsets of $\{1,...,n\}$. So if $M$ is fixed, your determinant is some polynomial in $a\_1,...,a\_n$ whose coefficients are principal minors of $M$. This is of course not difficult, but to continue, I would need to know what information about the determinant you want to get.


**Update.** By the way the same proof of course gives that coefficients of the characteristic polynomial of $M$ are (up to sign) sums of principal minors of $M$. This is a very useful fact, used, for instance, in one of the proofs of the classification of finite dimensional semi-simple Lie algebras (existence of Cartan subalgebras). 


