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Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincare are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.

Bounding Inequalities for Eigenvalues of Principal Submatrices