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The derivative of ann-order matrix function on the complex field is usually defined as an2{n}^{2}-order matrix, which is not suitable for generalizing Cauchy integral formula of matrix functions to its higher-order derivative form. In this paper, a new kind of derivative of matrix functions is defined, and the higher-order derivative form of Cauchy integral formula of matrix functions is also proved to be true under the new kind of definition of derivative. At the same time, some examples about calculating the values of matrix functions by using Cauchy integral formula of matrix functions and its higher-order derivative form are given.

Derivative and higher-order Cauchy integral formula of matrix functions