@article{Rabanovich_2020, title={Decomposition of a Hermitian matrix into a sum of a fixed number of orthoprojections}, volume={72}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/2378}, DOI={10.37863/umzh.v72i5.2378}, abstractNote={<p class="p1">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.</p>}, number={5}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={RabanovichV. I.}, year={2020}, month={Apr.}, pages={679–693} }