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

[math]\displaystyle{ \mathcal{K}(\mathbf{A})=\sup _{|z|\gt 1}(|z|-1)\left\|(z-\mathbf{A})^{-1}\right\|, }[/math]

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

[math]\displaystyle{ \mathcal{K}_{\textrm{lhp}}(\mathbf{A})=\sup _{\Re(z)\gt 0}(\Re(z))\left\|(z-\mathbf{A})^{-1}\right\|. }[/math]

Properties

• For any matrix A, one has that 𝒦(A) ≥ 1 and 𝒦_lhp(A) ≥ 1. In particular, 𝒦(A) (resp. 𝒦_lhp(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_lhp(A) = 1.
• 𝒦(A) and 𝒦_lhp(A) have alternative definitions through the pseudospectrum Λ_ε(A):
  • [math]\displaystyle{ \mathcal{K}(A)=\sup _{\varepsilon\gt 0} \frac{\rho_{\varepsilon}(A)-1}{\varepsilon} }[/math] , where p_ε(A) = max{|λ| : λ ∈ Λ_ε(A)},
  • [math]\displaystyle{ \mathcal{K}_{\textrm{lhp}}(A)=\sup _{\varepsilon\gt 0} \frac{\alpha_{\varepsilon}(A)}{\varepsilon} }[/math], where α_ε(A) = max{Re|λ| : λ ∈ Λ_ε(A)}.
• 𝒦_lhp(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

[math]\displaystyle{ \mathcal{K}(\mathbf{A}) \leq \sup_{k \geq 0}\left\|\mathbf{A}^k\right\| \leq e\, n\, \mathcal{K}(\mathbf{A}), }[/math]

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:

[math]\displaystyle{ \mathcal{K}_{\mathrm{lhp}}(\mathbf{A}) \leq \sup _{t \geq 0}\left\|\mathrm{e}^{t \mathbf{A}}\right\| \leq e \, n \, \mathcal{K}_{\mathrm{lhp}}(\mathbf{A}) }[/math]

Consequences and applications

The value [math]\displaystyle{ \sup_{k \geq 0}\left\|\mathbf{A}^k\right\| }[/math] (respectively, [math]\displaystyle{ \sup _{t \geq 0}\left\|\mathrm{e}^{t \mathbf{A}}\right\| }[/math]) can be interpreted as the maximum transient growth of the discrete-time system [math]\displaystyle{ x_{k+1}=A x_k }[/math] (respectively, continuous-time system [math]\displaystyle{ \dot{x}=A x }[/math]).

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.