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Maximum Principles in matrix theory

Published online by Cambridge University Press:  18 May 2009

L. Mirsky
L. Mirsky
Affiliation:
The University Sheffield
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Unless the contrary is stated, all matrices are understood to be complex and of type n × n. The transposed conjugate of A is denoted by A*. The non-negative square roots of the characteristic roots of A*A are called the singular values of A; they will be denoted by st(A), i = 1, …, n, where s1(A)≥…≥ sn(A). The symbol [A]k denotes the k × k submatrix standing in the upper left-hand corner of A. We shall write Ei(z1, …, zn) for the j-th elementary symmetric function of z1..., zn, and E1(A) for the j-th elementary symmetric function of the characteristic roots of A. It is understood that, throughout, 1≥j≥k≥n.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 1 , December 1958 , pp. 34 - 37
Copyright
Copyright © Glasgow Mathematical Journal Trust 1958

References

1.de Bruijn, N. G., Inequalities concerning minors and eigenvalues, Nieuw Archief v. Wiskunde (3), 4 (1956), 18–35.Google Scholar
2.Fan, Ky, On a theorem of Weyl concerning eigenvalues of linear transformations (I), Proc. Nat. Acad. Sci., 35 (1949), 652–655.Google Scholar
3.Fan, Ky, On a theorem of Weyl concerning eigenvalues of linear transformations (II), Proc. Nat. Acad. Sci., 36 (1950), 31–35.Google Scholar
4.Fan, Ky, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci., 37 (1951), 760–766.Google Scholar
5.Horn, A., On the singular values of a product of completely continuous operators, Proc. Nat. Acad. Sci., 36 (1950), 374–375.Google Scholar
6.Horn, A., On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc., 5 (1954), 4–7.Google Scholar
7.Marcus, M. and Moyls, B. N., On the maximum principle of Ky Fan, Ganad. J. Math., 9 (1957), 313–320.Google Scholar
8.Visser, C. and Zaanen, A. C., On the eigenvalues of compact linear transformations, Proc. Kon. Ned. Akad. Wetensch., Ser. A, 14 (1952), 71–78.Google Scholar
9.Weyl, H., Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci., 35 (1949), 408–411.Google Scholar