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] Given a matrix A, the Kreiss constant 𝒦(A) (with respect to the closed unit circle) of A is defined as[3]
sup
z
|
(
while the Kreiss constant 𝒦lhp(A) with respect to the left-half plane is given by[3]
lhp
{\displaystyle {\mathcal {K}}_{\textrm {lhp}}(\mathbf {A} )=\sup _{\Re (z)>0}(\Re (z))\left\|(z-\mathbf {A} )^{-1}\right\|.}
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]
{\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} )}
(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.