Kreiss matrix theorem

In matrix analysis, Kreiss matrix theorem relates the so-called Kreiss constant of a matrix with the power iterates of this matrix. It was originally introduced by Heinz-Otto Kreiss to analyze the stability of finite difference methods for partial difference equations.

Kreiss constant of a matrix

Given a matrix A, the Kreiss constant 𝒦(A) (with respect to the closed unit circle) of A is defined as

while the Kreiss constant 𝒦(A) with respect to the left-half plane is given by

Properties

• For any matrix A, one has that 𝒦(A) ≥ 1 and 𝒦(A) ≥ 1. In particular, 𝒦(A) (resp. 𝒦(A)) are finite only if the matrix A is Schur stable (resp. Hurwitz stable).
• Kreiss constant can be interpreted as a measure of normality of a matrix. In particular, for normal matrices A with spectral radius less than 1, one has that 𝒦(A) = 1. Similarly, for normal matrices A that are Hurwitz stable, 𝒦(A) = 1.
• 𝒦(A) and 𝒦(A) have alternative definitions through the pseudospectrum Λ(A):
• , where p(A) = max