Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that $0<q_{j}\leq 1$.

For which $A$ and $q$ there exists symmetric matrix $C$ such that \begin{align*} A\left\{ \frac{q_{j}}{\langle Ca_{j},a_{j}\rangle}\right\}A^{T}C=I_{k\times k} \end{align*} so that $\langle Ca_{j},a_{j}\rangle >0$ (here $\langle\cdot, \cdot\rangle$ denotes scalar product in Euclidian space), and $\left\{ \frac{q_{j}}{\langle Ca_{j},a_{j}\rangle}\right\}$ denotes $n\times n$ diagonal matrix.