Abstract
AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.
Highlights
The graph of an n-by-n Hermitian matrix M = has vertex set {, . . . , n} and edge set {ij | i < j, mij ≠ }
We prove that an n-by-n complex positive semide nite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero o -diagonal entries have modulus at least one, has trace at least n + r −
As a consequence, is the smallest limit point of the absolute trace of such matrices
Summary
Abstract: We prove that an n-by-n complex positive semide nite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero o -diagonal entries have modulus at least one, has trace at least n + r − . McKee and Pavlo Yatsyna [3] proved that an n-by-n positive de nite matrix S whose entries are integers and whose graph is connected must have trace at least n − .
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