Summary: The problem of finding a similarity scaling of a prespecifed structure to minimize the scaled singular value of a matrix arises frequently in the robustness analysis of control systems. Two questions that arise in this context are: (i) When is the set of optimal scalings nonempty? (ii) When is the optimal scaling unique? Sufficient conditions that guarantee an affirmative answer to questions (i) and (ii) are are available in [1], for the case when the scaling matrices are diagonal; a complete answer to these questions remains an open issue. The case when the scaling matrices are block-diagonal is essentially open.

References:

1. V. Balakrishnan and S. Boyd, ``Existence and uniqueness of optimal matrix scalings,'' SIAM J. on Matrix Analysis and Applications, vol. 16, pp. 29--39 (1995).