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Some generalizations of positive definiteness and monotonicity

✍ Scribed by Miroslav Fiedler; Vlastimil Pták

Publisher
Springer-Verlag
Year
1966
Tongue
English
Weight
775 KB
Volume
9
Category
Article
ISSN
0029-599X

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📜 SIMILAR VOLUMES

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Let \ be a nonnegative homogeneous function on R n . General structure of the set of numerical pairs ($, \*), for which the function (1&\ \* (x)) $ + is positive definite on R n is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic function

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For a coinmutative senugoup (S, +, \*) with involution and a function f : S 4 [O, m), the set S ( f ) of those p 2 0 such that f\* is a positive definite function on S is a closed subsemigroup of [O, 00) containing 0. For S = (Hi, +, G\* = -G) it may happen that S(f) = { kd : k E No } for some d>O,a

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In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.