Abstract
It is shown that the ratio of the area of the convex hull of the fields of values of the ( n−1)-by-( n−1) principal submatrices of an n-by- n matrix A to the area of the field of values of A is bounded below by a function of n which approaches 1 as n approaches ∞. Since this convex hull is necessarily contained in the field of values of A, an interpretation is that, asymptotically in the dimension, the field of any given matrix is “filled up” by the fields of the submatrices (collectively). Some new inequalities for the eigenvalues of principal submatrices of hermitian matrices, which are not implied by interlacing, are employed.
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