We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Dalecki\u{i}-Kre\u{i}n formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fr\'echet derivatives. Block forms of complex step approximations are introduced and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.

翻译：本文提出了形如$f(A(x))$的矩阵函数的一般偏导数理论，其中$A(x)$是多个变量（$x=(x_1,\dots,x_j)$）的矩阵路径。基于Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] 关于一阶导数的结果，我们发展了高阶偏导数的块上三角形式。该块形式被用于推导导数存在的条件以及高阶导数的广义Dalecki\u{i}-Kre\u{i}n公式。我们证明该公式的某些特例可导出量子微扰理论的经典公式。同时展示了我们的结果如何与已有的高阶Fréchet导数结果相关联。引入复步长逼近的块形式，并阐明其如何通过上三角形式与导数计算相关联。通过数值算例验证了这些关系。