We prove that any Hermitian matrix whose trace is integer and all eigenvalues lie in the segment [1 + 1/(k − 3),k − 1 − 1/(k − 3)] can be represented as a sum of k orthogonal projections. For the sums of k orthogonal projections, it is shown that the ratio of the number of eigenvalues that do not exceed 1 to the number of eigenvalues that are not smaller than 1 (with regard for the multiplicities) is not greater than k − 1. We also present examples of Hermitian matrices that satisfy the indicated condition for the numbers of eigenvalues but, at the same time, cannot be decomposed into the sum of k orthogonal projections.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 679–693, May, 2020.
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Rabanovich, V.I. Decomposition of a Hermitian Matrix into a Sum of Fixed Number of Orthogonal Projections. Ukr Math J 72, 785–802 (2020). https://doi.org/10.1007/s11253-020-01822-w
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DOI: https://doi.org/10.1007/s11253-020-01822-w