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On ranges of Lyapunov transformations IV†

Published online by Cambridge University Press:  18 May 2009

Raphael Loewy
Raphael Loewy
Affiliation:
Mathematics Research Center, University of Wisconsin-Madison
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Let ℂn,n denote the space of n × n matrices with complex entries and let ℋn denote the set of n × n hermitian matrices. Given any matrix A∊ℂn,n, the Lyapunov transformation corresponding to A is defined by ℐA(H) = AH+HA*, where H∊ℋn. Let PSD(n) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raised the problems of determining

and

Both of these problems seem to be difficult.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 2 , July 1976 , pp. 112 - 118
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

REFERENCES

1.Ben-Israel, A., Linear equations and inequalities of finite dimensional, real or complex, vector spaces: A unified theory, J. Math. Anal. Appl. 27 (1969), 367–389.CrossRefGoogle Scholar
2.Givens, W., Elementary divisors and some properties of the Lyapunov mapping X→AX+XA *, Argonne Nat. Lab. Rep. ANL–6456 (1961).Google Scholar
3.Loewy, R., On the Lyapunov transformation for stable matrices, Ph.D. thesis, California Institute of Technology (1972).Google Scholar
4.Loewy, R., On ranges of Lyapunov transformations III, SIAM J. Appl. Math. (to appear).Google Scholar
5.MacDuffee, C. C., The theory of matrices (Chelsea, 1956).Google Scholar
6.Rockafellar, R. T., Convex analysis (Princeton University Press, 1972).Google Scholar
7.Schneider, H., Positive operators and an inertia theorem, Numer. Math. 7 (1965), 11–17.CrossRefGoogle Scholar
8.Taussky, O., Matrix theory research problem, Bull. Amer. Math. Soc. 71 (1965), 711.Google Scholar
9.Taussky, O., Positive definite matrices, Inequalities (Shisha, O., editor) (Academic Press, 1967).Google Scholar
10.Taussky, O. and Wielandt, H., On the matrix function AX + X′A′, Arch. Rational Mech. Anal. 9 (1962), 93–96.CrossRefGoogle Scholar