mathematics

Quadratic eigenvalue problem

Quadratic eigenvalue problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quadratic eigenvalue problem rather than just read about it. In short: In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,} where Q ( λ ) = λ 2 M + λ C + K {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K} , with matrix coeffic…

Key takeaways

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

Reference excerpt

Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,}

where Q ( λ ) = λ 2 M + λ C + K {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K} , with matrix coefficients M , C , K ∈ C n × n {\displaystyle M,\,C,K\in \mathbb {C} ^{n\times n}} and we require that M ≠ 0 {\displaystyle M\,\neq 0} , (so that we have a nonzero leading coefficient). There are 2 n {\displaystyle 2n} eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. Q ( λ ) {\displaystyle Q(\lambda )} is also known as a quadratic polynomial matrix.

Spectral theory A QEP is said to be regular if det ( Q ( λ ) ) ≢ 0 {\displaystyle {\text{det}}(Q(\lambda ))\not \equiv 0} identically. The coefficient of the λ 2 n {\displaystyle \lambda ^{2n}} term in det ( Q ( λ ) ) {\displaystyle {\text{det}}(Q(\lambda ))} is det ( M ) {\displaystyle {\text{det}}(M)} , implying that the QEP is regular if M {\displaystyle M} is nonsingular. Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, λ 2 Q ( λ − 1 ) = λ 2 K + λ C + M {\displaystyle \lambda ^{2}Q(\lambda ^{-1})=\lambda ^{2}K+\lambda C+M} . As there are 2 n {\displaystyle 2n} eigenvectors in a n {\displaystyle n} dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.

Applications

Systems of differential equations Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:

M q ″ ( t ) + C q ′ ( t ) + K q ( t ) = 0 {\displaystyle Mq''(t)+Cq'(t)+Kq(t)=0}

Where q ( t ) ∈ R n {\displaystyle q(t)\in \mathbb {R} ^{n}} , and M , C , K ∈ R n × n {\displaystyle M,C,K\in \mathbb {R} ^{n\times n}} . If all quadratic eigenvalues of Q ( λ ) = λ 2 M + λ C + K {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K} are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as

q ( t ) = ∑ j = 1 2 n α j x j e λ j t = X e Λ t α {\displaystyle q(t)=\sum _{j=1}^{2n}\alpha _{j}x_{j}e^{\lambda _{j}t}=Xe^{\Lambda t}\alpha }

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Quadratic eigenvalue problem

Start with the simplest possible case. Write down what Quadratic eigenvalue problem 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 Quadratic eigenvalue problem 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 Quadratic eigenvalue problem 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 Quadratic eigenvalue problem

In research: Quadratic eigenvalue problem 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 Quadratic eigenvalue problem 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: Quadratic eigenvalue problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Applied mathematics stubs, Linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Quadratic eigenvalue problem 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.

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How to study Quadratic eigenvalue problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quadratic eigenvalue problem 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 Quadratic eigenvalue problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quadratic eigenvalue problem in simple terms?

In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\…

Why does Quadratic eigenvalue problem 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 Quadratic eigenvalue problem?

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 Quadratic eigenvalue problem.