# Partial Derivative of Trace of Matrix in negative power wrt to parameters

$$\renewcommand{\v}{\mathrm{vec}\left(#1\right)} \renewcommand{\m}{\mathbf{#1}} \renewcommand{\trace}{\mathrm{trace}\left(#1\right)} \renewcommand{\diag}{\mathrm{diag}\left(#1\right)}$$

Suppose we have the diagonal matrix $$\mathbf{D} = diag(\mathbf{HW1})$$, where $$W$$ is diagonal and known and $$1$$ is a column vector with ones. $$A$$ and $$B$$ are known matrices, also.

How can we calculate the partial derivative of the following wrt matrix calculus? $$\frac{\partial \trace{\mathbf A \mathbf D^{-1/2} \mathbf B)}}{\partial\m H}$$

This can be written as $$\frac{\partial \trace{\mathbf B\mathbf A \mathbf D^{-1/2} }}{\partial\m H} = \frac{\partial \trace{\mathbf{S} \mathbf D^{-1/2} }}{\partial\m H} = \trace {\frac{\partial {\mathbf{S} \mathbf D^{-1/2}} }{\partial\m H}}$$

Except for the analytical way, I couldn't find out in matrix cookbook something that can show me the way.

Define the operation $$B={\rm Diag}(b)$$ which takes a vector and returns a diagonal matrix,
and the operation $$b={\rm diag}(B)$$ which extracts the diagonal of a matrix into a vector.

Since $$W$$ is a diagonal matrix we can write it as $$W={\rm Diag}(w)$$ for some vector $$w$$.
The result of multiplying this matrix by a column of ones is simply $$\,w=W1$$.

Define some new variables \eqalign{ S &= BA, \quad &s &= {\rm diag}(S) = {\rm diag}(S^T) \cr g &= Hw, \quad &G &= {\rm Diag}(g), \quad dg = dH\,w \cr } Write the function of interest in terms of these new variables.
Then calculate its differential and its gradient. \eqalign{ \phi &= {\rm Tr}(SG^{-1/2}) = S^T:G^{-1/2} = s:g^{-1/2} \cr d\phi &= s:dg^{-1/2} \cr &= s:(-\tfrac{1}{2}g^{-3/2}\odot dg) \cr &= -\tfrac{1}{2}s:G^{-3/2}dg \cr &= -\tfrac{1}{2}G^{-3/2}s:dH\,w \cr &= -\tfrac{1}{2}G^{-3/2}sw^T:dH \cr \frac{\partial\phi}{\partial H} &= -\tfrac{1}{2}G^{-3/2}sw^T \cr &= -\tfrac{1}{2}{\rm Diag}(HW1)^{-3/2}\,{\rm diag}(BA)\,1^TW \cr } where functions on vectors are applied elementwise,
and $$(\odot)$$ represents the elementwise/Hadamard product,
and $$(\,:\,)$$ represents the trace/Frobenius product, i.e. $$\,A:B={\rm Tr}(A^TB)$$.

NB:  The cyclic property of the trace allows Frobenius products to be rearranged in numerous ways.
For example \eqalign{ A:BC &= B^TA:C \cr&= AC^T:B \cr&= BC:A \cr&= A^T:(BC)^T \cr } are all equivalent.

• Greg, your answer is incredible and thank you very much! I have a question. Which property did you use in order :dH to appear as a denumerator of partial \phi? (almost at the end of the main part of the solution). Mar 19, 2019 at 9:36
• Do you think I can follow the way you showed in order to solve $$trace ({\frac{\partial {\mathbf{S} \mathbf D^{-1}} }{\partial H}}), where\, \mathbf{D} = Diag(\mathbf{1}^{T} \mathbf{H})$$ ? Mar 19, 2019 at 11:28
• The trace is only defined for square matrices (and scalars if treated as $1\times 1$ matrices). Unfortunately, the quantity $$\frac{\partial(SD^{-1})}{\partial H}$$ is not a matrix, but rather a fourth-order tensor. For the second question, if it is known that the differential of a function is given by $$df = G\,dx$$ for any value of $dx$, then the quantity $G$ must be the gradient of the function. There is no division taking place, rather an identification that the only quantity having this property is the gradient. And the gradient is often written using Leibniz notation.
– greg
Mar 19, 2019 at 17:54
• Thank you for your help, Greg! Mar 20, 2019 at 10:19