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Bounds for Characteristic Values of Positive Definite Matrices

Published online by Cambridge University Press:  20 November 2018

P. A. Binding ,
W. D. Hoskins  and
P. J. Ponzo
P. A. Binding
Affiliation:
University of Manitoba, Winnipeg, Manitoba
W. D. Hoskins
Affiliation:
University of Manitoba, Winnipeg, Manitoba
P. J. Ponzo
Affiliation:
University of Waterloo, Waterloo, Ontario
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We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 51 - 56
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bodewig, , Matrix calculus, North Holland, Amsterdam (1965), 69-70.Google Scholar
2. Zhong-Ci, Shi and Bo-Ying, Wang, Bounds for the determinant, characteristic roots and condition number of certain types of matrices, Chinese Math. 7 (1965), 21-40.Google Scholar
3. Cargo, and Shisha, , Bounds on ratios of means, J. Res. Nat. Bur. Standards Vol. 66B (1962), 169-170.Google Scholar
4. Mond, and Shisha, , Inequalities, Vol. II, Academic Press, New York (1970), 241-249.Google Scholar