Comment #2 on issue 2759 by [email protected]: Implement the forumlas from the matrix cookbook
http://code.google.com/p/sympy/issues/detail?id=2759
First you have to understand what it means to take a derivative with respect to a vector. First, a vector depending on another vector (like y(x)) means that each component of y is a function of all the components of x. Thus, dy/dx is naturally defined as the Jacobian matrix (see D.3 from http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.AppD.d/IFEM.AppD.pdf). Notice that if y == A*x, then each component of dy/dx will be the same as A.T, because the way that the Jacobian is defined, each row correspond to the independent variables and the columns correspond to the dependent variable. This is the source of the transposition in all the formulas.
I think it may be possible to reduce the rules to the repeated application of simple rules, like the chain rule, etc., just like for normal derivatives, though I'm not 100% sure how to do it at this time.
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