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.

翻译：本短文证明，伴随矩阵 $\mbox{Adj}(z-A)$ 的高阶导数与矩阵 $A$ 的Jordan分解中的幂零矩阵和投影算子相关。这些关系表现为伴随矩阵的导数可分解为与特征值、幂零矩阵和投影算子相关的因子的乘积。利用Riesz投影算子和泛函演算获得了新的关系式。本文结果可视为Thompson-McEnteggert定理的推广，该定理将伴随矩阵与对称矩阵单特征值特征空间上的正交投影联系起来。这些结果也可视为对B. Parisse和M. Vaughan先前关于伴随矩阵导数与特征值不变子空间关系结果的补充。本文结果还可解释为广义特征向量-特征值恒等式。尽管已有诸多研究探讨了特征空间投影算子、伴随矩阵导数与特征空间之间的关系，但文献中似乎尚未明确提及Jordan分解定理中出现的幂零矩阵和投影矩阵参与的高阶伴随矩阵导数因子分解。