Is it possible to rewrite $X^* A X = B$ in ${\operatorname{vec}}(X)$, where $X \in \mathbb{C}^{n \times m}$, $A \in \mathbb{C}^{n \times n}$? Is it possible to vectorize the quadratic $$X^* A X = B$$ in ${\operatorname{vec}}(X)$, where $X \in \mathbb{C}^{n \times m}$, $A \in \mathbb{C}^{n \times n}$, and $B \in \mathbb{C}^{m \times m}$?


Apologies if it is trivial. I just don't know how to proceed. Your help will be highly appreciated.



I thought of using Kroncker Product. But it can represent only one side of the quadratic equation, i.e., either $X^*$ or $X$.



Partial attempt:

${\operatorname{vec}} (X^* A X) = \left( I \otimes X^* A \right) {\operatorname{vec}} (X) = {\operatorname{vec}} (B)$, then how to pull out $X^*$ from the Kronecker product? I want to represent the quadratic form in only ${\operatorname{vec}} (X^*)$ and ${\operatorname{vec}} (X)$.
See Wikipedia Vectorization Page.
See Wikipedia Vectorization Page.