Properties of determinant exponent

Question

https://en.wikipedia.org/wiki/Determinant#Properties_of_the_determinant lists some basic properties of a determinant:

$\det(A^T)=\det(A)$

$\det(AB)=\det(A)\det(B)$

$\det(cA)=c^n\det(A)$

What are the properties of: $\det(A^c)$?

Answer 1

Well, we have that for $k\in \mathbf{N}$ $$ \det(A^k)=\det(\underbrace{A\cdot A\cdots A}_{k\:\text{times}})=\det(A)^k$$ by applying the property $\det(AB)=\det(A)\det(B)$ inductively.

Answer 2

It's already been noted that it is easy to check for $k\in\mathbb{N}$ that $$\det(A^k)=\det(A)^k$$

We can do a little better though (and extend this to arbitrary $c\in\mathbb{C}$). I will rely on the use of the Matrix Exponential and Matrix Logarithm. These have definitely been discussed elsewhere on stackexchange, so I won't go into detail. The idea is, with numbers, we could write $$a^c=e^{c\ln(a)}$$

Given the power series definitions of matrix exponential and logarithm, it is clear that the same idea should work here. We can write $$A^c=e^{c\ln(A)}$$. Now, it is a known result that $$\det(e^A)=e^{\text{tr}(A)}$$ and $$\text{tr}(\ln(A))=\ln(\det(A))$$ Putting these together, we obtain $$\det(A^c)=\det(e^{c\ln(A)})=e^{\text{tr}(c\ln(A))}=e^{c\text{tr}(\ln(A))}=e^{c\ln(\det(A))}=\det(A)^c$$

Thus, the formula we derived for natural numbers holds in general (if the matrix exponent and logarithm are well defined - otherwise, we cannot make sense of $A^c$)