What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? ... oh sorry, "monicized derivative"?

--no operation, if M' and M both are the same size, say NxN [since degree(P')+1=degree(P)=N]. So your only hope is some sort of operation that removes a row & column (among other things). If we restricted attention to matrices of "companion" form https://en.wikipedia.org/wiki/Companion_matrix then the answer would be "remove the 1st row and column, then multiply the entries of the final column by 1/N,2/N,3/N,4/N,...,(N-1)/N." Which of course fails for transposed companion form etc. But anyhow, that tells us that the following admittedly somewhat contrived operation works for any NxN matrix M: 1. Express M = A C A^(-1) where C is of companion form. [I believe it is possible to arrange for this factorization to be unique if certain restrictions are placed on A...] 2. Zero the 1st row and column of C, then multiply the entries of the final column of the not-zeroed (N-1)x(N-1) block by 1/N,2/N,3/N,4/N,...,(N-1)/N. Result is C'. 3. M' = A C' A^(-1) has charpoly which is the monicized derivative of charpoly(M)... except times x so there is an extra zero eigenvalue.