In this paper we present an explicit construction for a representation, as a sum of orthoprojections, of a positive definite diagonal matrix with an integer trace exceeding its size. The existence of such a representation has been discovered by Fillmore without indicating a precise construction. We also describe scalar operators for which the above representation is unique (up to unitary equivalence). Moreover, we present another proof (see Theorem 2) of a result obtained earlier by Djeldubaev and Kruglyak.

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