3
$\begingroup$

Given that $A\in\mathbb{R}^{n\times n}$, is there an explicit matrix expression for

$$\frac{d}{dA}\log^2(\det(A))$$

(in numerator layout format)?

Attempt

$$=2\log(\det(A))\frac{d}{dA}\log(\det(A))$$ $$=2\log(\det(A))\frac{1}{\det(A)}\frac{d}{dA}\det(A)$$

$\endgroup$
5
  • $\begingroup$ Good work. Did you google "derivative of determinant"? You will find your answer if you do... $\endgroup$
    – runway44
    May 20, 2020 at 8:06
  • $\begingroup$ @runway44 I found Jacobi's formula but I don't know how to get the $dA$ out of the $\rm Tr()$ $\endgroup$
    – Museful
    May 20, 2020 at 8:08
  • 1
    $\begingroup$ The second equation in Wikipedia's "Jacobi formula" gives a "special case" which is precisely $\frac{d}{dA}\det A$, no traces involved. It's the transpose of the adjugate matrix (so, the cofactor matrix). $\endgroup$
    – runway44
    May 20, 2020 at 8:09
  • $\begingroup$ @runway44 I see... In numerator layout format would it be $adj(A)$ or $adj^T(A)$? $\endgroup$
    – Museful
    May 20, 2020 at 8:12
  • 2
    $\begingroup$ You can simplify $\frac{1}{\det(A)} \operatorname{adj}(A) = A^{-1}$. (Maybe needs transposing depending on layout convention) $\endgroup$
    – Hyperplane
    May 20, 2020 at 8:18

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.