!Hey there :)
I am currently working on a topic in control engineering and I'm currently looking for some way to relate determinants of matrix combinations to the determinant of the elements. Specifically, I would like to ask:
If I have the determinant of the hermitian matrix $A_H = AA^*$ (where $A$ is any complex n$\times$n matrix, and $A^*$ is the conjugate transpose), is it possible to draw conclusions about the determinant of $A$?
And similarly: If I have the determinant of the hermitian matrix $A_H = A+A^*$, is it possible to draw conclusions about the determinant of $A$?
Any hints are much appreciated, I hope I did not leave out anything important! Cheers and thanks, Christoph
This is a partial answer.
For $A \in \mathcal{M}_{n}(\mathbb{C})$,
$$\mathrm{det}(AA^{\ast})=\vert \mathrm{det}(A) \vert^{2}.$$
Using Schur's theorem, we know that there exists an unitary matrix $U$ such that and an upper triangular matrix :
$$ A = U T U^{\ast} $$
As a consequence, $A^{\ast} = UT^{\ast}U^{\ast}$ and $A+A^{\ast}=U \big( T+T^{\ast} \big) U^{\ast}$. It follows that :
$$ \mathrm{det}(A+A^{\ast}) = \prod \limits_{\lambda \in \mathrm{Sp}(A)} 2\Re(\lambda) $$
While $\displaystyle \mathrm{det}(A) = \prod \limits_{\lambda \in \mathrm{Sp}(A)} \lambda$.
In both cases, knowing $\mathrm{det}(AA^{\ast})$ or $\mathrm{det}(A+A^{\ast})$ is not enough to know $\mathrm{det}(A)$.