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

$𝒦(𝐀)=\underset{|z|>1}{\sup }(|z|-1)\Vert (z-𝐀{)}^{-1}\Vert ,$

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

${𝒦}_{\text{lhp}}(𝐀)=\underset{\mathrm{\Re }(z)>0}{\sup }(\mathrm{\Re }(z))\Vert (z-𝐀{)}^{-1}\Vert .$

Properties

• For any matrix A, one has that 𝒦(A) ≥ 1 and 𝒦Template:Sub(A) ≥ 1. In particular, 𝒦(A) (resp. 𝒦Template:Sub(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, 𝒦Template:Sub(A) = 1.
• 𝒦(A) and 𝒦Template:Sub(A) have alternative definitions through the pseudospectrum ΛTemplate:Sub(A):
  • $𝒦(A)=\underset{\epsilon >0}{\sup }\frac{{\rho }_{\epsilon }(A)-1}{\epsilon }$ , where pTemplate:Sub(A) = max{|λ| : λ ∈ ΛTemplate:Sub(A)},
  • ${𝒦}_{\text{lhp}}(A)=\underset{\epsilon >0}{\sup }\frac{{\alpha }_{\epsilon }(A)}{\epsilon }$, where αTemplate:Sub(A) = max{Re|λ| : λ ∈ ΛTemplate:Sub(A)}.
• 𝒦Template:Sub(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

$𝒦(𝐀)\le \underset{k\ge 0}{\sup }\Vert {𝐀}^k\Vert \le en𝒦(𝐀),$

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:

${𝒦}_{\mathrm{l}\mathrm{h}\mathrm{p}}(𝐀)\le \underset{t\ge 0}{\sup }\Vert {\mathrm{e}}^{t𝐀}\Vert \le en{𝒦}_{\mathrm{l}\mathrm{h}\mathrm{p}}(𝐀)$

Consequences and applications

The value $\underset{k\ge 0}{\sup }\Vert {𝐀}^k\Vert$ (respectively, $\underset{t\ge 0}{\sup }\Vert {\mathrm{e}}^{t𝐀}\Vert$) can be interpreted as the maximum transient growth of the discrete-time system $x_{k+1}=A{x}_k$ (respectively, continuous-time system $\dot{x}=Ax$).

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.