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Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. (English) Zbl 0585.62114
Summary: Several definitions are in use for the derivative of an $m×p$ matrix function F(X) with respect to its $n×q$ matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of $n×q$ matrices to the space of $m×p$ matrices it is often more convenient to study the map from nq-spaces to mp-spaces. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.
##### MSC:
 62H99 Multivariate analysis 26B12 Calculus of vector functions