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]

while the Kreiss constant š’¦lhp(A) with respect to the left-half plane is given by[3]

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 Ī›Īµ(A):[5]
    • , where pĪµ(A) = max{|Ī»| : Ī» āˆˆ Ī›Īµ(A)},
    • , where Ī±Īµ(A) = max{Re|Ī»| : Ī» āˆˆ Ī›Īµ(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]

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]

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.[5][6]

References

  1. ā†‘ Kreiss, Heinz-Otto (1962). "Ɯber Die StabilitƤtsdefinition FĆ¼r Differenzengleichungen Die Partielle Differentialgleichungen Approximieren". BIT. 2 (3): 153ā€“181. doi:10.1007/bf01957330. ISSN 0006-3835. S2CID 118346536.
  2. 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): 339ā€“360. doi:10.4064/-38-1-339-360. ISSN 0137-6934.
  3. 1 2 3 4 Raouafi, Samir (2018). "A generalization of the Kreiss Matrix Theorem". Linear Algebra and Its Applications. 549: 86ā€“99. doi:10.1016/j.laa.2018.03.011. S2CID 126237400.
  4. ā†‘ Jacob Nathaniel Stroh (2006). Non-normality in scalar delay differential equations (PDF) (Thesis).
  5. 1 2 Mitchell, Tim (2020). "Computing the Kreiss Constant of a Matrix". SIAM Journal on Matrix Analysis and Applications. 41 (4): 1944ā€“1975. arXiv:1907.06537. doi:10.1137/19m1275127. ISSN 0895-4798. S2CID 196622538.
  6. 1 2 Apkarian, Pierre; Noll, Dominikus (2020). "Optimizing the Kreiss Constant". SIAM Journal on Control and Optimization. 58 (6): 3342ā€“3362. arXiv:1910.12572. doi:10.1137/19m1296215. ISSN 0363-0129. S2CID 204904802.
  7. ā†‘ Trefethen, Lloyd N.; Embree, Mark (2005), Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, p. 177
  8. 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. hdl:1813/7113. JSTOR 2324361.
  9. 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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