  In this short note, we show that the higher-order derivatives of the adjugate
matrix $\mbox{Adj}(z-A)$, are related to the nilpotent matrices and projections
in the Jordan decomposition of the matrix $A$. These relations appear as a
factorization of the derivative of the adjugate matrix as a product of factors
related to the eigenvalues, nilpotent matrices and projectors. The novel
relations are obtained using the Riesz projector and functional calculus. The
results presented here can be considered to be a generalization of Thompson and
McEnteggert's theorem relating the adjugate matrix to the orthogonal projection
on the eigenspace of simple eigenvalues for symmetric matrices. They can also
be seen as a complement to some earlier results by B. Parisse, M. Vaughan that
relate derivatives of the adjugate matrix to the invariant subspaces associated
with an eigenvalue. Our results can also be interpreted as a general
eigenvector-eigenvalue identity. Many previous works have dealt with relations
between the projectors on the eigenspaces and the derivatives of the adjugate
matrix with the characteristic spaces but it seems that there is no explicit
mention in the literature of the factorization of the higher-order derivatives
of the adjugate matrix as a matrix multiplication involving nilpotent and
projector matrices, which appear in the Jordan decomposition theorem.
