The Eigenvalues of Complementary Principal Submatrices of a Positive Definite Matrix

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Abstract
Let C be an n-square Hermitian matrix, presented in partitioned form as where A is a-square and B is b-square. Let denote the eigenvalues of C, A, B, respectively. In a recent paper [10] the following inequality was established: 1.1 if 1.2 This inequality is a simplification and a sharpening of an inequality established earlier in [6], and is a wide generalization of an inequality of Aronszajn [4].

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