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 š¦(A) (with respect to the closed unit circle) of A is defined as[3]
[math]\displaystyle{ \mathcal{K}(\mathbf{A})=\sup _{|z|\gt 1}(|z|-1)\left\|(z-\mathbf{A})^{-1}\right\|, }[/math]
while the Kreiss constant š¦lhp(A) with respect to the left-half plane is given by[3]
[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 š¦(A) ā„ 1 and š¦lhp(A) ā„ 1. In particular, š¦(A) (resp. š¦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 š¦(A) = 1. Similarly, for normal matrices A that are Hurwitz stable, š¦lhp(A) = 1.
- š¦(A) and š¦lhp(A) have alternative definitions through the pseudospectrum ĪĪµ(A):[5]
- [math]\displaystyle{ \mathcal{K}(A)=\sup _{\varepsilon\gt 0} \frac{\rho_{\varepsilon}(A)-1}{\varepsilon} }[/math] , where pĪµ(A) = max{|Ī»| : Ī» ā ĪĪµ(A)},
- [math]\displaystyle{ \mathcal{K}_{\textrm{lhp}}(A)=\sup _{\varepsilon\gt 0} \frac{\alpha_{\varepsilon}(A)}{\varepsilon} }[/math], where Ī±Īµ(A) = max{Re|Ī»| : Ī» ā ĪĪµ(A)}.
- š¦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]
[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.[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]
[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.[5][6]
References
- ā Kreiss, Heinz-Otto (1962). "Ćber Die StabilitƤtsdefinition FĆ¼r Differenzengleichungen Die Partielle Differentialgleichungen Approximieren". BIT 2 (3): 153ā181. doi:10.1007/bf01957330. ISSN 0006-3835. http://dx.doi.org/10.1007/bf01957330.
- ā Strikwerda, John; Wade, Bruce (1997). "A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions". Banach Center Publications 38 (1): 339ā360. doi:10.4064/-38-1-339-360. ISSN 0137-6934. http://dx.doi.org/10.4064/-38-1-339-360.
- ā 3.0 3.1 3.2 3.3 Raouafi, Samir (2018). "A generalization of the Kreiss Matrix Theorem" (in en). Linear Algebra and Its Applications 549: 86ā99. doi:10.1016/j.laa.2018.03.011. https://linkinghub.elsevier.com/retrieve/pii/S0024379518301216.
- ā Jacob Nathaniel Stroh (2006). Non-normality in scalar delay differential equations (PDF) (Thesis).
- ā 5.0 5.1 Mitchell, Tim (2020). "Computing the Kreiss Constant of a Matrix". SIAM Journal on Matrix Analysis and Applications 41 (4): 1944ā1975. doi:10.1137/19m1275127. ISSN 0895-4798. http://dx.doi.org/10.1137/19m1275127.
- ā 6.0 6.1 Apkarian, Pierre; Noll, Dominikus (2020). "Optimizing the Kreiss Constant". SIAM Journal on Control and Optimization 58 (6): 3342ā3362. doi:10.1137/19m1296215. ISSN 0363-0129. http://dx.doi.org/10.1137/19m1296215.
- ā Trefethen, Lloyd N.; Embree, Mark (2005), Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, p. 177
- ā Wegert, Elias; Trefethen, Lloyd N. (1994). "From the Buffon Needle Problem to the Kreiss Matrix Theorem". The American Mathematical Monthly 101 (2): 132. doi:10.2307/2324361. https://www.jstor.org/stable/2324361.
- ā Trefethen, Lloyd N.; Embree, Mark (2005), Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, p. 183
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