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{λ| : λ ∈ Λ(A)}, , where α(A) = max{Re|λ| : λ ∈ Λ(A)}. 𝒦(A) can be computed through robust control methods.
Statement of Kreiss matrix theorem Let A be a square matrix of order n and e be the Euler's number. The modern and sharp version of Kreiss matrix theorem states that the inequality below is tight
and it follows from the application of Spijker's lemma.
There also exists an analogous result in terms of the Kreiss constant with respect to the left-half plane and the matrix exponential:
Consequences and applications The value (respectively, ) can be interpreted as the maximum transient growth of the discrete-time system (respectively, continuous-time system ).
Thus, the Kreiss matrix theorem gives both upper and lower bounds on the transient behavior of the system with dynamics given by the matrix A: a large (and finite) Kreiss constant indicates that the system will have an accentuated transient phase before decaying to zero.
References