- Source: Kreiss matrix theorem
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
K
(
A
)
=
sup
|
z
|
>
1
(
|
z
|
−
1
)
‖
(
z
−
A
)
−
1
‖
,
{\displaystyle {\mathcal {K}}(\mathbf {A} )=\sup _{|z|>1}(|z|-1)\left\|(z-\mathbf {A} )^{-1}\right\|,}
while the Kreiss constant 𝒦lhp(A) with respect to the left-half plane is given by
K
lhp
(
A
)
=
sup
ℜ
(
z
)
>
0
(
ℜ
(
z
)
)
‖
(
z
−
A
)
−
1
‖
.
{\displaystyle {\mathcal {K}}_{\textrm {lhp}}(\mathbf {A} )=\sup _{\Re (z)>0}(\Re (z))\left\|(z-\mathbf {A} )^{-1}\right\|.}
= Properties
=For any matrix A, one has that 𝒦(A) ≥ 1 and 𝒦lhp(A) ≥ 1. In particular, 𝒦(A) (resp. 𝒦lhp(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. 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, 𝒦lhp(A) = 1.
𝒦(A) and 𝒦lhp(A) have alternative definitions through the pseudospectrum Λε(A):
K
(
A
)
=
sup
ε
>
0
ρ
ε
(
A
)
−
1
ε
{\displaystyle {\mathcal {K}}(A)=\sup _{\varepsilon >0}{\frac {\rho _{\varepsilon }(A)-1}{\varepsilon }}}
, where pε(A) = max{|λ| : λ ∈ Λε(A)},
K
lhp
(
A
)
=
sup
ε
>
0
α
ε
(
A
)
ε
{\displaystyle {\mathcal {K}}_{\textrm {lhp}}(A)=\sup _{\varepsilon >0}{\frac {\alpha _{\varepsilon }(A)}{\varepsilon }}}
, where αε(A) = max{Re|λ| : λ ∈ Λε(A)}.
𝒦lhp(A) can be computed through robust control methods.
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
K
(
A
)
≤
sup
k
≥
0
‖
A
k
‖
≤
e
n
K
(
A
)
,
{\displaystyle {\mathcal {K}}(\mathbf {A} )\leq \sup _{k\geq 0}\left\|\mathbf {A} ^{k}\right\|\leq e\,n\,{\mathcal {K}}(\mathbf {A} ),}
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:
K
l
h
p
(
A
)
≤
sup
t
≥
0
‖
e
t
A
‖
≤
e
n
K
l
h
p
(
A
)
{\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} )}
Consequences and applications
The value
sup
k
≥
0
‖
A
k
‖
{\displaystyle \sup _{k\geq 0}\left\|\mathbf {A} ^{k}\right\|}
(respectively,
sup
t
≥
0
‖
e
t
A
‖
{\displaystyle \sup _{t\geq 0}\left\|\mathrm {e} ^{t\mathbf {A} }\right\|}
) can be interpreted as the maximum transient growth of the discrete-time system
x
k
+
1
=
A
x
k
{\displaystyle x_{k+1}=Ax_{k}}
(respectively, continuous-time system
x
˙
=
A
x
{\displaystyle {\dot {x}}=Ax}
).
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.