The Kreiss Matrix Theorem on a General Complex Domain

The Kreiss Matrix Theorem on a

On the Resolvent Condition I in the Kreiss

Abstrac t . The Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the constants in these two conditions grows linearly with N, and we obtain the optimal proportionality factor up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentials e At...

MATRIX VALUATION PSEUDO RING (MVPR) AND AN EXTENSION THEOREM OF MATRIX VALUATION

Let R be a ring and V be a matrix valuation on R. It is shown that, there exists a correspondence between matrix valuations on R and some special subsets ?(MVPR) of the set of all square matrices over R, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. Furthermore, based on Malcolmson’s localization, an alternative proof for t...

The Equivalence of L,StabiIity, the Resolvent Condition, and Strict H-Stability

The Kreiss matrix theorem asserts that a family of N X N matrices is L,-stable if and only if either a resolvent condition (R) or a Hennitian norm condition (H) is satisfied. We give a direct, considerahly shorter proof of the power-houndedness of an N X N matrix satisfying (R), sharpening former results by showing that powerhoundedness depends, at most, linearly on the dimension M. We also sho...

On the solving of matrix equation of Sylvester type

A solution of two problems related to the matrix equation of Sylvester type is given. In the first problem, the procedures for linear matrix inequalities are used to construct the solution of this equation. In the second problem, when a matrix is given which is not a solution of this equation, it is required to find such solution of the original equation, which most accurately approximates the ...