Kreiss matrix theorem explained

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 (A) (with respect to the closed unit circle) of A is defined as^[3]

l{K}( A )=\sup _{|z|>1} (|z| -1 )\left\|(z - A ) ^{-1} \right\|,

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

l{K} _{rm{lhp} }(\mathbf)=\sup _(\Re(z))\left\|(z-\mathbf)^\right\|.

Properties

• For any matrix A, one has that (A) ≥ 1 and (A) ≥ 1. In particular, (A) (resp. (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 (A) = 1. Similarly, for normal matrices A that are Hurwitz stable, (A) = 1.
• (A) and (A) have alternative definitions through the pseudospectrum Λ(A):^[5]

l{K}(A)=\sup _{\varepsilon>0} (\rho _{\varepsilon} (A) -1)/(\varepsilon) , where p(A) = max

λ ∈ Λ(A)

, l{K} _{rm{lhp} }(A)=\sup _ \frac, where α(A) = max.

• (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

l{K}( A ) \leq \sup _{k} \left\| A ^{k\right\|} \leq e n l{K}( A ),

and it follows from the application of Spijker's lemma.^[7]

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

l{K} _{lhp} ( A ) \leq \sup _{t} \left\| e ^{t} \right\| \leq e n l{K} _{lhp} ( A )

Consequences and applications

The value

\sup _{k} \left\| A ^{k\right\|} (respectively, \sup _{t} \left\| e ^{t} \right\| ) can be interpreted as the maximum transient growth of the discrete-time system x _{k+1} = A x _{k} (respectively, continuous-time system • x = A x ).

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.

Notes and References

1. Kreiss . Heinz-Otto . 1962 . Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren . BIT . 2 . 3 . 153–181 . 10.1007/bf01957330 . 118346536 . 0006-3835.
2. 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 . 339–360 . 10.4064/-38-1-339-360 . 0137-6934. free .
3. Raouafi . Samir . 2018 . A generalization of the Kreiss Matrix Theorem . Linear Algebra and Its Applications . en . 549 . 86–99 . 10.1016/j.laa.2018.03.011. 126237400 . free .
4. Non-normality in scalar delay differential equations . Jacob Nathaniel Stroh . 2006 .
5. Mitchell . Tim . 2020 . Computing the Kreiss Constant of a Matrix . SIAM Journal on Matrix Analysis and Applications . 41 . 4 . 1944–1975 . 10.1137/19m1275127 . 1907.06537 . 196622538 . 0895-4798.
6. Apkarian . Pierre . Noll . Dominikus . 2020 . Optimizing the Kreiss Constant . SIAM Journal on Control and Optimization . 58 . 6 . 3342–3362 . 1910.12572 . 10.1137/19m1296215 . 204904802 . 0363-0129.
7. Wegert . Elias . Trefethen . Lloyd N. . 1994 . From the Buffon Needle Problem to the Kreiss Matrix Theorem . The American Mathematical Monthly . 101 . 2 . 132 . 10.2307/2324361. 2324361 . 1813/7113 . free .