On the resolvent condition in the Kreiss Matrix Theorem

The Kreiss Matrix Theorem asserts the uniform equivalence over allN ×N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the constants in these two conditions grows linearly withN, and we obtain the optimal proportionality factor up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentialse^At. The proofs make use of a lemma that may be of independent interest, which bounds the arc length of the image of a circle in the complex plane under a rational function.Dedicated to Germund Dahlquist on the occasion of his sixtieth birthday.Research supported by NSF Mathematical Sciences Postdoctoral Fellowships, by the Courant Institute of Mathematical Sciences, and by the National Aeronautics and Space Administration under Contract No. NAS1-17070 while the authors were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665.