Decomposition of a Hermitian matrix into a sum of a fixed number of orthoprojections

Keywords: Orthoprojection, Hermitian matrix, Horn inequlities, Frame.

Abstract

We prove that any Hermitian matrix, whose trace is integer and all eigenvalues lie in $[1+1/(k-3),k-1-1/(k-3)],$ is a sum of $k$ orthoprojections. For sums of $k$ orthoprojections, it is shown that the ratio of the number of eigenvalues not exceeding 1 to the number of eigenvalues not less than 1, taking into account the multiplicity, is not greater than $k-1$. Examples of Hermitian matrices that satisfy the ratio for eigenvalues and, at the same time, can not be decomposed into a sum of $k$ orthoprojections are also suggested.

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Published
29.04.2020
How to Cite
RabanovichV. I. “Decomposition of a Hermitian Matrix into a Sum of a Fixed Number of Orthoprojections”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Apr. 2020, pp. 679–693, doi:10.37863/umzh.v72i5.2378.
Section
Research articles