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Can some one tell me how to get following matrix derivative or atleast point to source where i can find?I have looked at matrix cookbook but it doesn't have anything relevant

$\frac{d}{dW}\log (\det(\sigma^2+WW^T)) $

$\frac{d}{dW}Tr ((\sigma^2 I+WW^T)^{-1}S) $ where S doesn't contain any $W$ term.

I have read some reference saying results for this can be obtained from Krzanowski and Marriott 1994, but i couldn't find online reference for this too..

manifold
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1 Answers1

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Define the matrices $$\eqalign{ X &= (\sigma^2I+WW^T) \,\,= X^T \cr Y &= X^{-1}S \cr }$$ Then the two functions are $$\eqalign{ \phi &= \log(\det(X)) \cr \psi &= {\rm Tr}(Y) \cr }$$ Formulas for these can be found in the cookbook. Use that information to write their differential in terms of $(dX, dY)$ then perform a change of variables to $dW.$

$$\eqalign{ \frac{\partial\phi}{\partial X} &= X^{-T} \cr d\phi &= X^{-T}:dX \cr &= X^{-T}:(W\,dW^T+dW\,W^T) \cr &= (X^{-1}+X^{-T}):dW\,W^T \cr &= (X^{-1}+X^{-T})W:dW \cr &= 2X^{-1}W:dW \cr \frac{\partial\phi}{\partial W} &= 2X^{-1}W \cr \cr \frac{\partial\psi}{\partial Y} &= I \cr d\psi &= I:dY \cr &= I:dX^{-1}S \cr &= S^T:dX^{-1} \cr &= -S^T:X^{-1}\,dX\,X^{-1} \cr &= -X^{-T}S^TX^{-T}:dX \cr &= -X^{-T}S^TX^{-T}:(W\,dW^T+dW\,W^T) \cr &= -(X^{-1}SX^{-1}+X^{-T}S^TX^{-T}):dW\,W^T \cr &= -(X^{-1}SX^{-1}+X^{-T}S^TX^{-T}):dW\,W^T \cr &= -X^{-1}(S+S^T)X^{-1}W:dW \cr \frac{\partial\psi}{\partial W} &= -X^{-1}(S+S^T)X^{-1}W \cr }$$
NB: In these calculations, colons are used as a convenient product notation for the trace, i.e. $$A:B = {\rm Tr}(A^TB)$$
greg
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