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.^[1][2]

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^[3]

${\displaystyle {\mathcal {K}}(\mathbf {A} )=\sup _{|z|>1}(|z|-1)\left\|(z-\mathbf {A} )^{-1}\right\|,}$

while the Kreiss constant 𝒦_lhp(A) with respect to the left-half plane is given by^[3]

${\displaystyle {\mathcal {K}}_{\textrm {lhp}}(\mathbf {A} )=\sup _{\Re (z)>0}(\Re (z))\left\|(z-\mathbf {A} )^{-1}\right\|.}$

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.^[4] 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):^[5]
  • ${\displaystyle {\mathcal {K}}(A)=\sup _{\varepsilon >0}{\frac {\rho _{\varepsilon }(A)-1}{\varepsilon }}}$ , where p_ε(A) = max{|λ| : λ ∈ Λ_ε(A)},
  • ${\displaystyle {\mathcal {K}}_{\textrm {lhp}}(A)=\sup _{\varepsilon >0}{\frac {\alpha _{\varepsilon }(A)}{\varepsilon }}}$, where α_ε(A) = max{Re|λ| : λ ∈ Λ_ε(A)}.
• 𝒦_lhp(A) can be computed through robust control methods.^[6]

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^[3][7]

${\displaystyle {\mathcal {K}}(\mathbf {A} )\leq \sup _{k\geq 0}\left\|\mathbf {A} ^{k}\right\|\leq e\,n\,{\mathcal {K}}(\mathbf {A} ),}$

and it follows from the application of Spijker's lemma.^[8]

There also exists an analogous result in terms of the Kreiss constant with respect to the left-half plane and the matrix exponential:^[3][9]

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

Consequences and applications

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

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.^[5][6]