Matrix identity involving trace

Question

I am asked to prove

$$\frac{\partial }{\partial \alpha}(\log(\det(C)))=tr\left(C^{-1}\frac{\partial C}{\partial \alpha}\right).$$

$C$ is an invertible matrix whose entries are functions of $\alpha$. I get that the left hand side is the sum of logs of the eigenvalues. I have no clue how to proceed with RHS.

This result is known as Jacobi's formula. Although the presentation is overly complicated, there's a proof in Wikipedia. — user1551, Dec 17 '15 at 14:38

score 0 · Answer 1 · edited Apr 13 '17 at 12:19

0

Using the chain rule one finds the derivative $[d_A(\log\det A)](H)=\operatorname{trace}(A^{-1}H)$. Using the chain rule a level deeper we have $$[d_t(\log\det A_t)](\tau)=\operatorname{trace}\left(A_t^{-1}\;dA_t(\tau)\right)$$ denote $A_t^\prime$ the matrix of $dA_t$ and take the auxiliary argument $\tau$ out of the trace, then you have for the derivative map itself: $$d_t(\log\det A_t)=\operatorname{trace}\left(A_t^{-1}\;A_t^\prime\right)$$

edited Apr 13 '17 at 12:19

Community

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answered Dec 18 '15 at 01:14

rych

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related? https://math.stackexchange.com/questions/1493137/prove-frac-partial-rmlnx-partial-x-2x-1-rmdiagx-1 – BCLC Apr 16 '21 at 10:22

Matrix identity involving trace

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