Higher order derivatives of the adjugate matrix and the Jordan form

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 a generalization of the Thompson and McEnteggert theorem that relates the adjugate matrix with the orthogonal projection on the eigenspace of simple eigenvalues for symmetric matrices. They can also be viewed as a complement to some previous results by B. Parisse, M. Vaughan that related derivatives of the adjugate matrix with 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 derivatives of the adjugate matrix with the characteristic spaces but it seems there is no explicit mention in the literature of the factorization of the higher-order derivatives of the adjugate matrix as a product involving nilpotent and projector matrices that appears in the Jordan decomposition theorem.