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Nonnegative definite Hermitian matrices with increasing principal minors. (English) Zbl 1291.15089

Summary: A nonnegative definite Hermitian \(m \times m\) matrix \(A\neq 0\) has increasing principal minors if \(\det A[I] \leqslant \det A[J]\) for \(I\subset J\), where \(\det A[I]\) is the principal minor of \(A\) based on rows and columns in the set \(I \subseteq \{1,\dots,m\}\). For \(m > 1\) we show \(A\) has increasing principal minors if and only if \(A^{-1}\) exists and its diagonal entries are less or equal to 1.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
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