I am trying to find a matrix $A$ that satisfies the following type of equation:
$$A^\dagger A = f(B,C,...)$$
where the right-hand side is a function of known matrices $B,C,...$
Is there a general approach for this?
EDIT: The matrices $B,C,\dots$ are fixed and independent, and the right-hand side does not depend on $A$.
We have the following matrix equation in $\mathrm X \in \mathbb C^{n \times n}$
$$\mathrm X^* \mathrm X = \mathrm A$$
where $\mathrm A \in \mathbb C^{n \times n}$ is given. Since $\mathrm X^* \mathrm X$ is Hermitian and positive semidefinite, $\mathrm A$ must also be Hermitian and positive semidefinite. Hence, $\mathrm A$ is diagonalizable. Let its eigendecomposition be
$$\mathrm A = \mathrm V \Lambda \mathrm V^*$$
Thus, one solution to the matrix equation is
$$\bar{\mathrm X} := \Lambda^{\frac 12} \mathrm V^*$$