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The truncated trigonometric matrix moment problem is a well studied object. In particular, one can find in the literature the extremal value concerning the weight assigned to some point of the open unit disk within the solution set of that matrix moment problem for the so-called nondegenerate situation and some special measure which realizes the extremal value. The primary concern of the paper is to give a basic proof for the fact that this distinguished measure is uniquely determined by that extremal feature.

On maximal weight solutions in a truncated trigonometric matrix moment problem