2 problems

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Conjecture on positivity-free rank characterizations for singular state matrices
Rank-characterization conjecture. When $A$ is singular, the LMI $$ should have the same solution set as the equation
Conjectureopenall-pass-functions discrete-time-systems linear-matrix-inequalitiesarXiv posted 2 days ago
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Conjecture on Lyapunov representations of observable all-pass functions
Lyapunov representation conjecture. There exists a symmetric matrix $P=P^{\top}$ satisfying
Conjectureopenall-pass-functions discrete-time-systems lyapunov-equationsarXiv posted 2 days ago

Conjecture on positivity-free rank characterizations for singular state matrices