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