The derivative of a matrix element with respect to the matrix:
Is there any equation or formula for it?
I could not find it on the Web. If you know any book treating it, I would appreciate the title.
By the chain rule, .
If that inverse does not exist, the derivative does not exist.
I agree with HallsofIvy. It is not orthodox mathematics to differentiate with respect to a matrix. Nevertheless, some people have tried to formulate such a concept, and you may find this Wikipedia page informative.
According to that page, if f(X) is a scalar-valued function of an n×m matrix X then the derivative is defined to be the m×n matrix whose (i,j)-element is . In particular, if then would be a matrix with a 1 in the (j,i)-position and zeros elsewhere.
But notice that much of the material on that Wikipedia page is disputed. I have no idea how reliable or useful this whole concept is.
I see it's better if I post a shorter version of the whole function.
I need to obtain the analytic formula of following derivative:
where the dxd matrix Sigma is positive definite and decomposed as
is the n-dimensional vector composed of the diagonal of the matrix Sigma.
Although not much relevant, where ai is the i-th row of A.
Thanks for any help.
P.S. Actually, like it could be that . Thus, when i=j. Zero when i<>j.