Kreiss matrix theorem

Given a matrix A, the Kreiss constant 𝒦(A) (with respect to the closed unit circle) of A is defined as[3]

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

while the Kreiss constant 𝒦_lhp(A) with respect to the left-half plane is given by[3]

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

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]

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

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

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

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

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.[5][6]