Given $$A_{i,j,k} X_j^* X_k + C_i = 0$$ where $𝐴_{i,j,k}$ and $𝐶_i$ are arbitrary complex numbers for all $𝑗, 𝑘$ which are $𝑁$-dimensional indices and $i$ which is an $m$-dimensional index where $m<N$. Note $*$ is complex conjugation and there is a summation for repeated indices. Are there conditions on $A$ and $C$ to guarantee the existence of a solution to this set of equations?
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$\begingroup$ One can formulate this in terns of Hermitian matrices $R_i$ and $S_i$ so that $A_i=R_i + \sqrt{-1} S_i$. The equations then become $X^t R_i X^* + D_i=0$ and $X^t S_i X^* + E_i=0$ where $C_i=D_i + \sqrt{-1} E_i$. Note that $D_i$ and $E_i$ are real numbers. $\endgroup$– KapilCommented May 10, 2021 at 5:10
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