Kreiss matrix theorem

Given a matrix A, the Kreiss constant 𝒦(A) (with respect to the closed unit circle) of A is defined as{{Cite journal |last=Raouafi |first=Samir |date=2018 |title=A generalization of the Kreiss Matrix Theorem |journal=Linear Algebra and Its Applications |language=en |volume=549 |pages=86–99 |doi=10.1016/j.laa.2018.03.011|s2cid=126237400 |doi-access=free }}

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

while the Kreiss constant 𝒦{{Sub|lhp}}(A) with respect to the left-half plane is given by

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

• For any matrix A, one has that 𝒦(A) ≥ 1 and 𝒦{{Sub|lhp}}(A) ≥ 1. In particular, 𝒦(A) (resp. 𝒦{{Sub|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.{{Cite thesis |title=Non-normality in scalar delay differential equations |author=Jacob Nathaniel Stroh |year=2006 |url=https://bueler.github.io/papers/jnstrohMS.pdf}} 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{{Sub|lhp}}(A) = 1.
• 𝒦(A) and 𝒦{{Sub|lhp}}(A) have alternative definitions through the pseudospectrum Λ{{Sub|ε}}(A):{{Cite journal |last=Mitchell |first=Tim |date=2020 |title=Computing the Kreiss Constant of a Matrix |url=http://dx.doi.org/10.1137/19m1275127 |journal=SIAM Journal on Matrix Analysis and Applications |volume=41 |issue=4 |pages=1944–1975 |doi=10.1137/19m1275127 |arxiv=1907.06537 |s2cid=196622538 |issn=0895-4798}}
• $\backslash mathcal\{K\}(A)=\backslash sup\; \_\{\backslash varepsilon>0\}\; \backslash frac\{\backslash rho\_\{\backslash varepsilon\}(A)-1\}\{\backslash varepsilon\}$ , where p{{Sub|ε}}(A) = max{|λ| : λ ∈ Λ{{Sub|ε}}(A)},
• $\backslash mathcal\{K\}\_\{\backslash textrm\{lhp\}\}(A)=\backslash sup\; \_\{\backslash varepsilon>0\}\; \backslash frac\{\backslash alpha\_\{\backslash varepsilon\}(A)\}\{\backslash varepsilon\}$, where α{{Sub|ε}}(A) = max{Re|λ| : λ ∈ Λ{{Sub|ε}}(A)}.
• 𝒦{{Sub|lhp}}(A) can be computed through robust control methods.{{Cite journal |last1=Apkarian |first1=Pierre |last2=Noll |first2=Dominikus |date=2020 |title=Optimizing the Kreiss Constant |url=http://dx.doi.org/10.1137/19m1296215 |journal=SIAM Journal on Control and Optimization |volume=58 |issue=6 |pages=3342–3362 |arxiv=1910.12572 |doi=10.1137/19m1296215 |s2cid=204904802 |issn=0363-0129}}

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{{Citation |last1=Trefethen |first1=Lloyd N. |title=Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators |page=177 |year=2005 |publisher=Princeton University Press |last2=Embree |first2=Mark}}

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

and it follows from the application of Spijker's lemma.{{Cite journal |last1=Wegert |first1=Elias |last2=Trefethen |first2=Lloyd N. |date=1994 |title=From the Buffon Needle Problem to the Kreiss Matrix Theorem |url=https://www.jstor.org/stable/2324361 |journal=The American Mathematical Monthly |volume=101 |issue=2 |pages=132 |doi=10.2307/2324361|jstor=2324361 |hdl=1813/7113 |hdl-access=free }}

There also exists an analogous result in terms of the Kreiss constant with respect to the left-half plane and the matrix exponential:{{Citation |last1=Trefethen |first1=Lloyd N. |title=Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators |page=183 |year=2005 |publisher=Princeton University Press |last2=Embree |first2=Mark}}

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

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

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.

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Category:Numerical linear algebra

Category:Spectral theory

Category:Systems theory

Category:Theorems in linear algebra