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Some inequalities concerning positive-definite Hermitian matrices

Published online by Cambridge University Press:  24 October 2008

Ky Fan
Affiliation:
The American University and University of Notre Dame

Extract

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Fan, K.On a theorem of Weyl concerning eigenvalues of linear transformations. II. Proc. nat. Acad. Sci., Wash., 36 (1950), 31–5.CrossRefGoogle ScholarPubMed
(2)Fan, K.Problem 4429. Amer. math. Mon. 58 (1951), 194; 60 (1953), 48–50.Google Scholar
(3)Fan, K. Inequalities for eigenvalues of Hermitian matrices. Contributions to the solution of systems of linear equations and the determination of eigenvalues (edited by Taussky, O., National Bureau of Standards Applied Mathematics Series, 1954), 131–9.Google Scholar
(4)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities (Cambridge, 1934).Google Scholar