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

Authors

DOI:

https://doi.org/10.37863/umzh.v72i5.2378

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.

References

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Published

29.04.2020

Issue

Vol. 72 No. 5 (2020)

Section

Research articles

How to Cite

Rabanovich, V. 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, https://doi.org/10.37863/umzh.v72i5.2378.