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.
Kreiss constant of a matrix.
Given a matrix "A", the Kreiss constant 𝒦("A") (with respect to the closed unit circle) of "A" is defined as
formula_1
while the Kreiss constant 𝒦lhp("A") with respect to the left-half plane is given by
formula_2
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
formula_5
and it follows from the application of Spijker's lemma.
There also exists an analogous result in terms of the Kreiss constant with respect to the left-half plane and the matrix exponential:
formula_6
Consequences and applications.
The value formula_7 (respectively, formula_8) can be interpreted as the "maximum transient growth" of the discrete-time system formula_9 (respectively, continuous-time system formula_10).
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.
References.
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