Eigenvalue interlacing for certain classes of matrices with real principal minors (Q1090387)

Let A be a complex square matrix. The authors study the eigenvalue interlacing property (every principal submatrix of A has a real eigenvalue, and the two smallest real eigenvalues of A are interlaced by the smallest real eigenvalue of every principal submatrix of order one less), the positive GLP property (either some generalized leading principal (GLP) minor of A is nonpositive, or all principal minors of A are positive), the class of \(\omega\)-matrices having the eigenvalue monotonicity property, and others. They show that A has the eigenvalue interlacing property iff A is an \(\omega\)-matrix and every principal submatrix x of A has the semipositive GLP property.