  A systematic theory is introduced for calculating the derivatives of
quaternion matrix function with respect to quaternion matrix variables. The
proposed methodology is equipped with the matrix product rule and chain rule
and it is able to handle both analytic and nonanalytic functions. This corrects
a flaw in the existing methods, that is, the incorrect use of the traditional
product rule. In the framework introduced, the derivatives of quaternion matrix
functions can be calculated directly without the differential of this function.
Key results are summarized in tables. Several examples show how the quaternion
matrix derivatives can be used as an important tool for solving problems
related to signal processing.
