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.

中文翻译：

---

将 Hermitian 矩阵分解为固定数量的正交投影的总和

我们证明任何迹为整数且所有特征值都位于 [1 + 1/(k − 3),k − 1 − 1/(k − 3)] 段内的 Hermitian 矩阵可以表示为 k 个正交投影的和. 对于k个正交投影的和，表明不超过1的特征值的数量与不小于1的特征值的数量（就多重性而言）的比值不大于k-1。我们还提供了 Hermitian 矩阵的例子，它满足特征值数量的指示条件，但同时不能分解为 k 个正交投影的总和。