Can we solve the following matrix equation in $K$
$$KQK^*=I$$
where $Q$ is non-singular and $I$ is the identity matrix, using SDP and LMI constraints?
2025-01-13 00:14:35.1736727275
Can we solve $KQK^*=I$?
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No. Nonlinear equality constraints are not convex, so there is no way to represent this using LMIs. Consider the $n=1$ case, $k^2q=1$; its solution set is $k=\pm 1/\sqrt{q}$.