no, it's not really new, as you can see here:

http://en.wikipedia.org/wiki/Matrix_calculus

http://en.wikipedia.org/wiki/Matrix_calculus#Derivatives_with_matrices

However, I am not sure, wether you can define the type of the result of dM/dX mathematically.

I can reccomend a good good book "The Matrix Cookbook" by

Kaare Brandt Petersen

Michael Syskind Pedersen

http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf

This textbook is very usefull to understand what already done in this field.

Good luck!

Hello. As far as I know, the derivative of a scalar, vector or matrix with respect to a matrix is not a new idea. A well known reference is Magnus and Neudecker's "Matrix differential calculus". I have also added a link to a paper where this type of method is used to analyse a nonlinear least squares algorithm. Some ideas are summarised in the appendix to this paper. The main trick is to make sure that basic operations are preserved by the differentiation rule (chain rule, etc.) so that more complicated problems can be tackled.

Article Analysis of a Nonlinear Least Squares Procedure Used in Glob...

Thanks to both, the paper and the book will be useful for me

Those techniques are currently used in thye matricial formulation of quantum mechanics;For instance,look at the Dirac textbook on QM...

The difficult became when matrix is not squared, in the example that I placed before it was solved by avoiding to divide by the non squared matrix, but it is not always possible.