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.

翻译：本文简短地展示了伴随矩阵 $\mbox{Adj}(z-A)$ 的高阶导数与矩阵 $A$ 的约当分解中的幂零矩阵和投影算子相关联。这些关系表现为伴随矩阵的导数可分解为与特征值、幂零矩阵及投影算子相关的因子乘积。利用Riesz投影与泛函演算，我们得到了这些新关系。本文结果可被视为Thompson和McEnteggert关于对称矩阵简单特征值对应的特征空间上正交投影伴随矩阵定理的推广，也可视为B. Parisse和M. Vaughan将伴随矩阵导数与特征值不变子空间关联的早期结果的补充。我们的结果还可解释为广义特征向量-特征值恒等式。尽管此前许多文献已涉及特征空间投影算子、伴随矩阵导数与特征空间之间的关系，但尚未见文献明确指出伴随矩阵的高阶导数可分解为约当分解定理中出现的幂零矩阵和投影矩阵的乘积形式。