Kreiss matrix theorem

Key takeaways

Kreiss matrix theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kreiss matrix theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kreiss matrix theorem from memory before moving on to harder problems.

Reference excerpt

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} )}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Kreiss matrix theorem

Start with the simplest possible case. Write down what Kreiss matrix theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kreiss matrix theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kreiss matrix theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kreiss matrix theorem

In research: Kreiss matrix theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kreiss matrix theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kreiss matrix theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical linear algebra, Spectral theory, Systems theory, so understanding it makes those chapters shorter.
In everyday life: Look for Kreiss matrix theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Kreiss matrix theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kreiss matrix theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kreiss matrix theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kreiss matrix theorem in simple terms?

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.

Why does Kreiss matrix theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kreiss matrix theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kreiss matrix theorem.