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Background

Jacobi's formula tells us that

$$\frac{d}{dt}\det A(t) = \operatorname{tr} \left( \operatorname{adj}(A(t)) \frac{dA(t)}{dt} \right) = (\det A(t)) \cdot \operatorname{tr} \left( A(t)^{-1} \cdot \frac{dA(t)}{dt} \right)$$

for the first derivative of a matrix $A(t)$ with respect to parameter $t$. A previous question showed that there exists such a formula

$$\frac{\partial^2}{\partial \alpha^2}\det A= \det(A) \left[\text{tr}^2\left( A^{-1} A_{\alpha} \right) + \text{tr} \left( A^{-1} A_{\alpha^2} \right)+N\right]$$

for the second partial derivatives.

Question

Is there such a formula for the $k$th order derivative

$$\frac{d^k}{dt^k}\det A(t) = ?$$

of the determinant of such a matrix?

Note that the question nth derivative of determinant wrt matrix is only a special case since here I am assuming each entry is function of a parameter $t$ that I am taking the derivative with respect to.

Galen
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  • Have you resolved this issue? – dtn Aug 21 '22 at 09:45
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    @dtn Not yet. In the mean time it suffices for me to use a computer algebra system for a given derivative order and matrix size. But an answer here would be appreciated! – Galen Aug 21 '22 at 13:08

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