Generalizations of two inequalities involving hermitian forms |
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Authors: | Sin-Chung Chang |
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Institution: | National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135, USA |
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Abstract: | Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, . Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products and , where M(y) is a matrix defined by . Subsequently, these results are extended to improve the spectral bounds of . |
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